1 l8.022 (E&M) -Lecture 2 Topics: Energy stored in a system of charges Electric field: concept and...
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Transcript of 1 l8.022 (E&M) -Lecture 2 Topics: Energy stored in a system of charges Electric field: concept and...
1
l8022 (EampM) -Lecture 2
Topics
Energy stored in a system of charges
Electric field concept and problems
Gaussrsquos law and its applications
Feedback
Thanks for the feedback
Scared by Pset 0 Almost all of the math used in the course is in ithellip
Math review too fast Will review new concepts again before using them
Pace ofectures too fast We have a lot to cover buthellip please remind me
2
Last timehellip
Coulombrsquos law
Superposition principle
September 8 2004 8022 ndash Lecture 1
3
Energy associated with FCoulomb
How much work do I have to do to move q from r1 to r2
September 8 2004 8022 ndash Lecture 1
4
Work done to move charges
Assuming radial path
How much work do I have to do to move q from r1 to r2
Does this result depend on the path chosen
No You can decompose any path in segments to the radial direction and segments |_ to it Since the component on the |_ is nul the result does not change
September 8 2004 8022 ndash Lecture 1
5
Corollaries
The work performed to move a charge between P1 and P2 is the same independently of the path chosen
The work to move a charge on a close path is zero
In other words the electrostatic force is conservative
This will allow us to introduce the concept of potential (next week)
6
Energy of a system of charges
How much work does it take to assemble a certain configuration of charges
Energy stored by N charges
September 8 2004 8022 ndash Lecture 1
7
The electric field
Q what is the best way of describing the effect of charges
1 charge in the Universe 1048708 2 charges in the Universe
But the force F depends on the test charge qhellip
define a quantity that describes the effect of the charge Q on the surroundings Electric Field
Units dynesesu
September 8 2004 8022 ndash Lecture 1
8
lElectric field lines
Visualize the direction and strength of the Electric Field
Direction to E pointing towards ndash and away from +
Magnitude the denser the lines the stronger the field
Properties
Field lines never cross (if so thatrsquo where E = 0)
They are orthogonal to equipotential surfaces (will see this later)
September 8 2004 8022 ndash Lecture 1
9
Electric field of a ring of charge
Problem Calculate the electric field created by a uniformly charged ring on its axis
Special case center of the ring General case any point P on the axis
Answers
bull Center of the ring E=0 by symmetry
bull General case
September 8 2004 8022 ndash Lecture 1
10
Electric field of disk of charge
Problem Find the electric field created by adisk of charges on the axis of the disk
Tricka disk is the sum of an infinite number of infinitely thin concentric rings And we know Eringhellip
(creative recycling is fair game in physics)
September 8 2004 8022 ndash Lecture 1
11
E of disk of charge (cont)
If charge is uniformly spread
Electric field created by the ring is
Integrating on r 01048708R
Electric field of a ring of radius r
September 8 2004 8022 ndash Lecture 1
12
Special case 1 R1048708infinity
For finite R
What if Rinfinity Eg what if Rgtgtz
Since
Conclusion
Electric Field created by an infinite conductive plane Direction perpendicular to the plane (+-z) Magnitude 2πσ (constant )
September 8 2004 8022 ndash Lecture 1
13
Special case 2 hgtgtR
For finite R
What happens when hgtgtR
Physicistrsquos approach The disk will look like a point charge with Q=σπ r2
Mathematicians approach Calculate from the previous result for zgtgtR (Taylor expansion)
14
The concept of flux
Consider the flow of water in a river The water velocity is described by
Immerse a squared wire loop of area A in the water (surface S)
Define the loop area vector as
Q how much water will flow through the loop Eg What is the ldquoflux of the velocityrdquo through the surface S
15
What is the flux of the velocity
It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop
General case (definition of flux)
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
2
Last timehellip
Coulombrsquos law
Superposition principle
September 8 2004 8022 ndash Lecture 1
3
Energy associated with FCoulomb
How much work do I have to do to move q from r1 to r2
September 8 2004 8022 ndash Lecture 1
4
Work done to move charges
Assuming radial path
How much work do I have to do to move q from r1 to r2
Does this result depend on the path chosen
No You can decompose any path in segments to the radial direction and segments |_ to it Since the component on the |_ is nul the result does not change
September 8 2004 8022 ndash Lecture 1
5
Corollaries
The work performed to move a charge between P1 and P2 is the same independently of the path chosen
The work to move a charge on a close path is zero
In other words the electrostatic force is conservative
This will allow us to introduce the concept of potential (next week)
6
Energy of a system of charges
How much work does it take to assemble a certain configuration of charges
Energy stored by N charges
September 8 2004 8022 ndash Lecture 1
7
The electric field
Q what is the best way of describing the effect of charges
1 charge in the Universe 1048708 2 charges in the Universe
But the force F depends on the test charge qhellip
define a quantity that describes the effect of the charge Q on the surroundings Electric Field
Units dynesesu
September 8 2004 8022 ndash Lecture 1
8
lElectric field lines
Visualize the direction and strength of the Electric Field
Direction to E pointing towards ndash and away from +
Magnitude the denser the lines the stronger the field
Properties
Field lines never cross (if so thatrsquo where E = 0)
They are orthogonal to equipotential surfaces (will see this later)
September 8 2004 8022 ndash Lecture 1
9
Electric field of a ring of charge
Problem Calculate the electric field created by a uniformly charged ring on its axis
Special case center of the ring General case any point P on the axis
Answers
bull Center of the ring E=0 by symmetry
bull General case
September 8 2004 8022 ndash Lecture 1
10
Electric field of disk of charge
Problem Find the electric field created by adisk of charges on the axis of the disk
Tricka disk is the sum of an infinite number of infinitely thin concentric rings And we know Eringhellip
(creative recycling is fair game in physics)
September 8 2004 8022 ndash Lecture 1
11
E of disk of charge (cont)
If charge is uniformly spread
Electric field created by the ring is
Integrating on r 01048708R
Electric field of a ring of radius r
September 8 2004 8022 ndash Lecture 1
12
Special case 1 R1048708infinity
For finite R
What if Rinfinity Eg what if Rgtgtz
Since
Conclusion
Electric Field created by an infinite conductive plane Direction perpendicular to the plane (+-z) Magnitude 2πσ (constant )
September 8 2004 8022 ndash Lecture 1
13
Special case 2 hgtgtR
For finite R
What happens when hgtgtR
Physicistrsquos approach The disk will look like a point charge with Q=σπ r2
Mathematicians approach Calculate from the previous result for zgtgtR (Taylor expansion)
14
The concept of flux
Consider the flow of water in a river The water velocity is described by
Immerse a squared wire loop of area A in the water (surface S)
Define the loop area vector as
Q how much water will flow through the loop Eg What is the ldquoflux of the velocityrdquo through the surface S
15
What is the flux of the velocity
It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop
General case (definition of flux)
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
3
Energy associated with FCoulomb
How much work do I have to do to move q from r1 to r2
September 8 2004 8022 ndash Lecture 1
4
Work done to move charges
Assuming radial path
How much work do I have to do to move q from r1 to r2
Does this result depend on the path chosen
No You can decompose any path in segments to the radial direction and segments |_ to it Since the component on the |_ is nul the result does not change
September 8 2004 8022 ndash Lecture 1
5
Corollaries
The work performed to move a charge between P1 and P2 is the same independently of the path chosen
The work to move a charge on a close path is zero
In other words the electrostatic force is conservative
This will allow us to introduce the concept of potential (next week)
6
Energy of a system of charges
How much work does it take to assemble a certain configuration of charges
Energy stored by N charges
September 8 2004 8022 ndash Lecture 1
7
The electric field
Q what is the best way of describing the effect of charges
1 charge in the Universe 1048708 2 charges in the Universe
But the force F depends on the test charge qhellip
define a quantity that describes the effect of the charge Q on the surroundings Electric Field
Units dynesesu
September 8 2004 8022 ndash Lecture 1
8
lElectric field lines
Visualize the direction and strength of the Electric Field
Direction to E pointing towards ndash and away from +
Magnitude the denser the lines the stronger the field
Properties
Field lines never cross (if so thatrsquo where E = 0)
They are orthogonal to equipotential surfaces (will see this later)
September 8 2004 8022 ndash Lecture 1
9
Electric field of a ring of charge
Problem Calculate the electric field created by a uniformly charged ring on its axis
Special case center of the ring General case any point P on the axis
Answers
bull Center of the ring E=0 by symmetry
bull General case
September 8 2004 8022 ndash Lecture 1
10
Electric field of disk of charge
Problem Find the electric field created by adisk of charges on the axis of the disk
Tricka disk is the sum of an infinite number of infinitely thin concentric rings And we know Eringhellip
(creative recycling is fair game in physics)
September 8 2004 8022 ndash Lecture 1
11
E of disk of charge (cont)
If charge is uniformly spread
Electric field created by the ring is
Integrating on r 01048708R
Electric field of a ring of radius r
September 8 2004 8022 ndash Lecture 1
12
Special case 1 R1048708infinity
For finite R
What if Rinfinity Eg what if Rgtgtz
Since
Conclusion
Electric Field created by an infinite conductive plane Direction perpendicular to the plane (+-z) Magnitude 2πσ (constant )
September 8 2004 8022 ndash Lecture 1
13
Special case 2 hgtgtR
For finite R
What happens when hgtgtR
Physicistrsquos approach The disk will look like a point charge with Q=σπ r2
Mathematicians approach Calculate from the previous result for zgtgtR (Taylor expansion)
14
The concept of flux
Consider the flow of water in a river The water velocity is described by
Immerse a squared wire loop of area A in the water (surface S)
Define the loop area vector as
Q how much water will flow through the loop Eg What is the ldquoflux of the velocityrdquo through the surface S
15
What is the flux of the velocity
It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop
General case (definition of flux)
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
4
Work done to move charges
Assuming radial path
How much work do I have to do to move q from r1 to r2
Does this result depend on the path chosen
No You can decompose any path in segments to the radial direction and segments |_ to it Since the component on the |_ is nul the result does not change
September 8 2004 8022 ndash Lecture 1
5
Corollaries
The work performed to move a charge between P1 and P2 is the same independently of the path chosen
The work to move a charge on a close path is zero
In other words the electrostatic force is conservative
This will allow us to introduce the concept of potential (next week)
6
Energy of a system of charges
How much work does it take to assemble a certain configuration of charges
Energy stored by N charges
September 8 2004 8022 ndash Lecture 1
7
The electric field
Q what is the best way of describing the effect of charges
1 charge in the Universe 1048708 2 charges in the Universe
But the force F depends on the test charge qhellip
define a quantity that describes the effect of the charge Q on the surroundings Electric Field
Units dynesesu
September 8 2004 8022 ndash Lecture 1
8
lElectric field lines
Visualize the direction and strength of the Electric Field
Direction to E pointing towards ndash and away from +
Magnitude the denser the lines the stronger the field
Properties
Field lines never cross (if so thatrsquo where E = 0)
They are orthogonal to equipotential surfaces (will see this later)
September 8 2004 8022 ndash Lecture 1
9
Electric field of a ring of charge
Problem Calculate the electric field created by a uniformly charged ring on its axis
Special case center of the ring General case any point P on the axis
Answers
bull Center of the ring E=0 by symmetry
bull General case
September 8 2004 8022 ndash Lecture 1
10
Electric field of disk of charge
Problem Find the electric field created by adisk of charges on the axis of the disk
Tricka disk is the sum of an infinite number of infinitely thin concentric rings And we know Eringhellip
(creative recycling is fair game in physics)
September 8 2004 8022 ndash Lecture 1
11
E of disk of charge (cont)
If charge is uniformly spread
Electric field created by the ring is
Integrating on r 01048708R
Electric field of a ring of radius r
September 8 2004 8022 ndash Lecture 1
12
Special case 1 R1048708infinity
For finite R
What if Rinfinity Eg what if Rgtgtz
Since
Conclusion
Electric Field created by an infinite conductive plane Direction perpendicular to the plane (+-z) Magnitude 2πσ (constant )
September 8 2004 8022 ndash Lecture 1
13
Special case 2 hgtgtR
For finite R
What happens when hgtgtR
Physicistrsquos approach The disk will look like a point charge with Q=σπ r2
Mathematicians approach Calculate from the previous result for zgtgtR (Taylor expansion)
14
The concept of flux
Consider the flow of water in a river The water velocity is described by
Immerse a squared wire loop of area A in the water (surface S)
Define the loop area vector as
Q how much water will flow through the loop Eg What is the ldquoflux of the velocityrdquo through the surface S
15
What is the flux of the velocity
It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop
General case (definition of flux)
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
5
Corollaries
The work performed to move a charge between P1 and P2 is the same independently of the path chosen
The work to move a charge on a close path is zero
In other words the electrostatic force is conservative
This will allow us to introduce the concept of potential (next week)
6
Energy of a system of charges
How much work does it take to assemble a certain configuration of charges
Energy stored by N charges
September 8 2004 8022 ndash Lecture 1
7
The electric field
Q what is the best way of describing the effect of charges
1 charge in the Universe 1048708 2 charges in the Universe
But the force F depends on the test charge qhellip
define a quantity that describes the effect of the charge Q on the surroundings Electric Field
Units dynesesu
September 8 2004 8022 ndash Lecture 1
8
lElectric field lines
Visualize the direction and strength of the Electric Field
Direction to E pointing towards ndash and away from +
Magnitude the denser the lines the stronger the field
Properties
Field lines never cross (if so thatrsquo where E = 0)
They are orthogonal to equipotential surfaces (will see this later)
September 8 2004 8022 ndash Lecture 1
9
Electric field of a ring of charge
Problem Calculate the electric field created by a uniformly charged ring on its axis
Special case center of the ring General case any point P on the axis
Answers
bull Center of the ring E=0 by symmetry
bull General case
September 8 2004 8022 ndash Lecture 1
10
Electric field of disk of charge
Problem Find the electric field created by adisk of charges on the axis of the disk
Tricka disk is the sum of an infinite number of infinitely thin concentric rings And we know Eringhellip
(creative recycling is fair game in physics)
September 8 2004 8022 ndash Lecture 1
11
E of disk of charge (cont)
If charge is uniformly spread
Electric field created by the ring is
Integrating on r 01048708R
Electric field of a ring of radius r
September 8 2004 8022 ndash Lecture 1
12
Special case 1 R1048708infinity
For finite R
What if Rinfinity Eg what if Rgtgtz
Since
Conclusion
Electric Field created by an infinite conductive plane Direction perpendicular to the plane (+-z) Magnitude 2πσ (constant )
September 8 2004 8022 ndash Lecture 1
13
Special case 2 hgtgtR
For finite R
What happens when hgtgtR
Physicistrsquos approach The disk will look like a point charge with Q=σπ r2
Mathematicians approach Calculate from the previous result for zgtgtR (Taylor expansion)
14
The concept of flux
Consider the flow of water in a river The water velocity is described by
Immerse a squared wire loop of area A in the water (surface S)
Define the loop area vector as
Q how much water will flow through the loop Eg What is the ldquoflux of the velocityrdquo through the surface S
15
What is the flux of the velocity
It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop
General case (definition of flux)
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
6
Energy of a system of charges
How much work does it take to assemble a certain configuration of charges
Energy stored by N charges
September 8 2004 8022 ndash Lecture 1
7
The electric field
Q what is the best way of describing the effect of charges
1 charge in the Universe 1048708 2 charges in the Universe
But the force F depends on the test charge qhellip
define a quantity that describes the effect of the charge Q on the surroundings Electric Field
Units dynesesu
September 8 2004 8022 ndash Lecture 1
8
lElectric field lines
Visualize the direction and strength of the Electric Field
Direction to E pointing towards ndash and away from +
Magnitude the denser the lines the stronger the field
Properties
Field lines never cross (if so thatrsquo where E = 0)
They are orthogonal to equipotential surfaces (will see this later)
September 8 2004 8022 ndash Lecture 1
9
Electric field of a ring of charge
Problem Calculate the electric field created by a uniformly charged ring on its axis
Special case center of the ring General case any point P on the axis
Answers
bull Center of the ring E=0 by symmetry
bull General case
September 8 2004 8022 ndash Lecture 1
10
Electric field of disk of charge
Problem Find the electric field created by adisk of charges on the axis of the disk
Tricka disk is the sum of an infinite number of infinitely thin concentric rings And we know Eringhellip
(creative recycling is fair game in physics)
September 8 2004 8022 ndash Lecture 1
11
E of disk of charge (cont)
If charge is uniformly spread
Electric field created by the ring is
Integrating on r 01048708R
Electric field of a ring of radius r
September 8 2004 8022 ndash Lecture 1
12
Special case 1 R1048708infinity
For finite R
What if Rinfinity Eg what if Rgtgtz
Since
Conclusion
Electric Field created by an infinite conductive plane Direction perpendicular to the plane (+-z) Magnitude 2πσ (constant )
September 8 2004 8022 ndash Lecture 1
13
Special case 2 hgtgtR
For finite R
What happens when hgtgtR
Physicistrsquos approach The disk will look like a point charge with Q=σπ r2
Mathematicians approach Calculate from the previous result for zgtgtR (Taylor expansion)
14
The concept of flux
Consider the flow of water in a river The water velocity is described by
Immerse a squared wire loop of area A in the water (surface S)
Define the loop area vector as
Q how much water will flow through the loop Eg What is the ldquoflux of the velocityrdquo through the surface S
15
What is the flux of the velocity
It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop
General case (definition of flux)
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
7
The electric field
Q what is the best way of describing the effect of charges
1 charge in the Universe 1048708 2 charges in the Universe
But the force F depends on the test charge qhellip
define a quantity that describes the effect of the charge Q on the surroundings Electric Field
Units dynesesu
September 8 2004 8022 ndash Lecture 1
8
lElectric field lines
Visualize the direction and strength of the Electric Field
Direction to E pointing towards ndash and away from +
Magnitude the denser the lines the stronger the field
Properties
Field lines never cross (if so thatrsquo where E = 0)
They are orthogonal to equipotential surfaces (will see this later)
September 8 2004 8022 ndash Lecture 1
9
Electric field of a ring of charge
Problem Calculate the electric field created by a uniformly charged ring on its axis
Special case center of the ring General case any point P on the axis
Answers
bull Center of the ring E=0 by symmetry
bull General case
September 8 2004 8022 ndash Lecture 1
10
Electric field of disk of charge
Problem Find the electric field created by adisk of charges on the axis of the disk
Tricka disk is the sum of an infinite number of infinitely thin concentric rings And we know Eringhellip
(creative recycling is fair game in physics)
September 8 2004 8022 ndash Lecture 1
11
E of disk of charge (cont)
If charge is uniformly spread
Electric field created by the ring is
Integrating on r 01048708R
Electric field of a ring of radius r
September 8 2004 8022 ndash Lecture 1
12
Special case 1 R1048708infinity
For finite R
What if Rinfinity Eg what if Rgtgtz
Since
Conclusion
Electric Field created by an infinite conductive plane Direction perpendicular to the plane (+-z) Magnitude 2πσ (constant )
September 8 2004 8022 ndash Lecture 1
13
Special case 2 hgtgtR
For finite R
What happens when hgtgtR
Physicistrsquos approach The disk will look like a point charge with Q=σπ r2
Mathematicians approach Calculate from the previous result for zgtgtR (Taylor expansion)
14
The concept of flux
Consider the flow of water in a river The water velocity is described by
Immerse a squared wire loop of area A in the water (surface S)
Define the loop area vector as
Q how much water will flow through the loop Eg What is the ldquoflux of the velocityrdquo through the surface S
15
What is the flux of the velocity
It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop
General case (definition of flux)
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
8
lElectric field lines
Visualize the direction and strength of the Electric Field
Direction to E pointing towards ndash and away from +
Magnitude the denser the lines the stronger the field
Properties
Field lines never cross (if so thatrsquo where E = 0)
They are orthogonal to equipotential surfaces (will see this later)
September 8 2004 8022 ndash Lecture 1
9
Electric field of a ring of charge
Problem Calculate the electric field created by a uniformly charged ring on its axis
Special case center of the ring General case any point P on the axis
Answers
bull Center of the ring E=0 by symmetry
bull General case
September 8 2004 8022 ndash Lecture 1
10
Electric field of disk of charge
Problem Find the electric field created by adisk of charges on the axis of the disk
Tricka disk is the sum of an infinite number of infinitely thin concentric rings And we know Eringhellip
(creative recycling is fair game in physics)
September 8 2004 8022 ndash Lecture 1
11
E of disk of charge (cont)
If charge is uniformly spread
Electric field created by the ring is
Integrating on r 01048708R
Electric field of a ring of radius r
September 8 2004 8022 ndash Lecture 1
12
Special case 1 R1048708infinity
For finite R
What if Rinfinity Eg what if Rgtgtz
Since
Conclusion
Electric Field created by an infinite conductive plane Direction perpendicular to the plane (+-z) Magnitude 2πσ (constant )
September 8 2004 8022 ndash Lecture 1
13
Special case 2 hgtgtR
For finite R
What happens when hgtgtR
Physicistrsquos approach The disk will look like a point charge with Q=σπ r2
Mathematicians approach Calculate from the previous result for zgtgtR (Taylor expansion)
14
The concept of flux
Consider the flow of water in a river The water velocity is described by
Immerse a squared wire loop of area A in the water (surface S)
Define the loop area vector as
Q how much water will flow through the loop Eg What is the ldquoflux of the velocityrdquo through the surface S
15
What is the flux of the velocity
It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop
General case (definition of flux)
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
9
Electric field of a ring of charge
Problem Calculate the electric field created by a uniformly charged ring on its axis
Special case center of the ring General case any point P on the axis
Answers
bull Center of the ring E=0 by symmetry
bull General case
September 8 2004 8022 ndash Lecture 1
10
Electric field of disk of charge
Problem Find the electric field created by adisk of charges on the axis of the disk
Tricka disk is the sum of an infinite number of infinitely thin concentric rings And we know Eringhellip
(creative recycling is fair game in physics)
September 8 2004 8022 ndash Lecture 1
11
E of disk of charge (cont)
If charge is uniformly spread
Electric field created by the ring is
Integrating on r 01048708R
Electric field of a ring of radius r
September 8 2004 8022 ndash Lecture 1
12
Special case 1 R1048708infinity
For finite R
What if Rinfinity Eg what if Rgtgtz
Since
Conclusion
Electric Field created by an infinite conductive plane Direction perpendicular to the plane (+-z) Magnitude 2πσ (constant )
September 8 2004 8022 ndash Lecture 1
13
Special case 2 hgtgtR
For finite R
What happens when hgtgtR
Physicistrsquos approach The disk will look like a point charge with Q=σπ r2
Mathematicians approach Calculate from the previous result for zgtgtR (Taylor expansion)
14
The concept of flux
Consider the flow of water in a river The water velocity is described by
Immerse a squared wire loop of area A in the water (surface S)
Define the loop area vector as
Q how much water will flow through the loop Eg What is the ldquoflux of the velocityrdquo through the surface S
15
What is the flux of the velocity
It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop
General case (definition of flux)
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
10
Electric field of disk of charge
Problem Find the electric field created by adisk of charges on the axis of the disk
Tricka disk is the sum of an infinite number of infinitely thin concentric rings And we know Eringhellip
(creative recycling is fair game in physics)
September 8 2004 8022 ndash Lecture 1
11
E of disk of charge (cont)
If charge is uniformly spread
Electric field created by the ring is
Integrating on r 01048708R
Electric field of a ring of radius r
September 8 2004 8022 ndash Lecture 1
12
Special case 1 R1048708infinity
For finite R
What if Rinfinity Eg what if Rgtgtz
Since
Conclusion
Electric Field created by an infinite conductive plane Direction perpendicular to the plane (+-z) Magnitude 2πσ (constant )
September 8 2004 8022 ndash Lecture 1
13
Special case 2 hgtgtR
For finite R
What happens when hgtgtR
Physicistrsquos approach The disk will look like a point charge with Q=σπ r2
Mathematicians approach Calculate from the previous result for zgtgtR (Taylor expansion)
14
The concept of flux
Consider the flow of water in a river The water velocity is described by
Immerse a squared wire loop of area A in the water (surface S)
Define the loop area vector as
Q how much water will flow through the loop Eg What is the ldquoflux of the velocityrdquo through the surface S
15
What is the flux of the velocity
It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop
General case (definition of flux)
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
11
E of disk of charge (cont)
If charge is uniformly spread
Electric field created by the ring is
Integrating on r 01048708R
Electric field of a ring of radius r
September 8 2004 8022 ndash Lecture 1
12
Special case 1 R1048708infinity
For finite R
What if Rinfinity Eg what if Rgtgtz
Since
Conclusion
Electric Field created by an infinite conductive plane Direction perpendicular to the plane (+-z) Magnitude 2πσ (constant )
September 8 2004 8022 ndash Lecture 1
13
Special case 2 hgtgtR
For finite R
What happens when hgtgtR
Physicistrsquos approach The disk will look like a point charge with Q=σπ r2
Mathematicians approach Calculate from the previous result for zgtgtR (Taylor expansion)
14
The concept of flux
Consider the flow of water in a river The water velocity is described by
Immerse a squared wire loop of area A in the water (surface S)
Define the loop area vector as
Q how much water will flow through the loop Eg What is the ldquoflux of the velocityrdquo through the surface S
15
What is the flux of the velocity
It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop
General case (definition of flux)
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
12
Special case 1 R1048708infinity
For finite R
What if Rinfinity Eg what if Rgtgtz
Since
Conclusion
Electric Field created by an infinite conductive plane Direction perpendicular to the plane (+-z) Magnitude 2πσ (constant )
September 8 2004 8022 ndash Lecture 1
13
Special case 2 hgtgtR
For finite R
What happens when hgtgtR
Physicistrsquos approach The disk will look like a point charge with Q=σπ r2
Mathematicians approach Calculate from the previous result for zgtgtR (Taylor expansion)
14
The concept of flux
Consider the flow of water in a river The water velocity is described by
Immerse a squared wire loop of area A in the water (surface S)
Define the loop area vector as
Q how much water will flow through the loop Eg What is the ldquoflux of the velocityrdquo through the surface S
15
What is the flux of the velocity
It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop
General case (definition of flux)
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
13
Special case 2 hgtgtR
For finite R
What happens when hgtgtR
Physicistrsquos approach The disk will look like a point charge with Q=σπ r2
Mathematicians approach Calculate from the previous result for zgtgtR (Taylor expansion)
14
The concept of flux
Consider the flow of water in a river The water velocity is described by
Immerse a squared wire loop of area A in the water (surface S)
Define the loop area vector as
Q how much water will flow through the loop Eg What is the ldquoflux of the velocityrdquo through the surface S
15
What is the flux of the velocity
It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop
General case (definition of flux)
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
14
The concept of flux
Consider the flow of water in a river The water velocity is described by
Immerse a squared wire loop of area A in the water (surface S)
Define the loop area vector as
Q how much water will flow through the loop Eg What is the ldquoflux of the velocityrdquo through the surface S
15
What is the flux of the velocity
It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop
General case (definition of flux)
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
15
What is the flux of the velocity
It depends on how the oop is oriented wrt the waterhellip Assuming constant velocity and plane loop
General case (definition of flux)
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
16
FAQ what is the direction of dA
Defined unambiguously only for a 3d surface At any point in space dA is perpendicular to the surface It points towards the ldquooutsiderdquo of the surface
Examples
Intuitively
ldquoda is oriented in such a way that if we have a hose inside the surface the flux through the surface will be positiverdquo
September 8 2004 8022 ndash Lecture 1
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
17
Flux of Electric Field
Definition
Example uniform electric field + flat surface
Calculate the flux
Interpretation Represent E using field lines ΦE is proportional to Nfield lines that go through the loop
NB this interpretation is valid for any electric field andor surface
September 8 2004 8022 ndash Lecture 1
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
18
ΦE through closed (3d) surface
Consider the total flux of E through a cylinder
Calculate
Cylinder axis is to field lines
The total flux through the cylinder is zero
because
but opposite sign since
September 8 2004 8022 ndash Lecture 1
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
19
ΦE through closed empty surface
Q1 Is this a coincidence due to shapeorientation of the cylinder
Clue
Think about interpretation of ΦE proportional of field lines through the surfacehellip
Answer
No all field lines that get into the surface have to come out
Conclusion
The electric flux through a closed surface that does not containcharges is zero
September 8 2004 8022 ndash Lecture 1
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
20
ΦE through surface containing Q
Q1 What if the surface contains charges
Clue
Think about interpretaton of ΦE the lines will ether originate in the surface (positive flux) or terminate inside the surface (negative flux)
Conclusion
The electric flux through a closed surface that does contain a netcharge is non zero
September 8 2004 8022 ndash Lecture 1
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
21
Simple example
ΦE of charge at center of sphere
Problem
Calculate ΦE for point charge +Q at the center of a sphere of radius R
Solution
everywhere on the sphere
Point charge at distance
September 8 2004 8022 ndash Lecture 1
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
22
ΦE through a generic surface
What if the surface is not spherical S
Impossible integral
Use intuition and interpretation of flux
Version 1
Consider the sphere S1
Field lines are always continuous
Version 2
Purcell 110 or next lecture
Conclusion
The electric flux Φ through any closed surface S containing a net charge Qis proportional to the charge enclosed
September 8 2004 8022 ndash Lecture 1
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
23
Thoughts on Gaussrsquos law
Why is Gaussrsquos law so important
Because it relates the electric field E with its sources Q
Given Q distribution find E (ntegral form) Given E find Q (differential form next week)
Is Gaussrsquos law always true Yes no matter what E or what S the flux is always = 4πQ
Is Gaussrsquos law always useful
No itrsquos useful only when the problem has symmetries
(Gaussrsquos law in integral form)
September 8 2004 8022 ndash Lecture 1
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
24
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Problem Calculate the electric field (everywhere in space) due to a spherical distribution of positive charges or radius R (NB solid sphere with volume charge density ρ)
Approach 1 (mathematician)
bull I know the E due to a pont charge dq dE=dqr2
bull I know how to integrate bull Sove the integralnsde and outsde the sphere (eg rltR and rgtR)
Comment correct but usually heavy on math
Approach 2 (physicist)
bull Why would I ever solve an ntegrals somebody (Gauss) already did it for me bull Just use Gaussrsquos theoremhellip
Comment correct much much less time consuming
September 8 2004 8022 ndash Lecture 1
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
25
Applications of Gaussrsquos law
Electric field of spherical distribution of charges
Physicistrsquos solution 1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
2) Inside the sphere (rltR) Apply Gauss on a sphere S2 of radius r
September 8 2004 8022 ndash Lecture 1
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
26
Do I get full credit for this solution
Did I answer the question completely
No I was asked to determine the electric field The electric field is a vector magnitude and direction
How to get the E direction
Look at the symmetry of the problem Spherical symmetry E must point radially
Complete solution
September 8 2004 8022 ndash Lecture 1
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
27
Another application of Gaussrsquos law
Electric field of spherical shell
Problem Calculate the electric field (everywhere in space) due to a positively charged spherical shell or radius R (surface charge density σ)
Physicistrsquos solutionapply Gauss
1) Outside the sphere (rgtR)
Apply Gauss on a sphere S1 of radius r
1) Inside the sphere (rltR)
Apply Gauss on a sphere S2 of radius r But sphere is hollow Qenclosed =0 E=0
NB spherical symmetry E is radial
September 8 2004 8022 ndash Lecture 1
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
28
Still another application of Gaussrsquos law
Electric field of infinite sheet of charge
Problem Calculate the electric feld at a distance z from a positively charged infinite plane of surface charge density σ
Again apply Gauss
Trick 1 choose the right Gaussian surface
Look at the symmetry of the problem Choose a cylinder of area A and height +-z
Trick 2 apply Gaussrsquos theorem
September 8 2004 8022 ndash Lecture 1
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
29
Checklist for solving 8022 problems
Read the problem (I am not jokng) Look at the symmetries before choosng the best coordinate system Look at the symmetries agan and find out what cancels what and the direction of the vectors nvolved
Look for a way to avoid all complicated integration
Remember physicists are lazy complicated integra you screwed up somewhere or there is an easier way out
Turn the math crankhellip Write down the compete solution (magnitudes and directions for all the different regions)
Box the solution your graders will love you If you encounter expansions
Find your expansion coefficient (xltlt1) and ldquomassagerdquo the result until you get something that looks like (1+x)N(1-x)N or ln(1+x) or ex
Donrsquot stop the expansion too early Taylor expansions are more than limitshellip
September 8 2004 8022 ndash Lecture 1
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-
30
Summary and outlook
What have we learned so far
Energy of a system of charges
Concept of electric field E
To describe the effect of charges independenty from the test charge
Gaussrsquos theorem in integral form
Useful to derive E from charge distributon with easy calculations
Next time
Derive Gaussrsquos theorem in a more rigorous way
See Purcel 110 if you cannot waithellip
Gaussrsquos law in differential form
hellip with some more intro to vector calculushellip
Useful to derive charge distributon given the electric felds
Energy associated with an electric field
September 8 2004 8022 ndash Lecture 1
- Slide 1
- Slide 2
- Slide 3
- Slide 4
- Slide 5
- Slide 6
- Slide 7
- Slide 8
- Slide 9
- Slide 10
- Slide 11
- Slide 12
- Slide 13
- Slide 14
- Slide 15
- Slide 16
- Slide 17
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Slide 22
- Slide 23
- Slide 24
- Slide 25
- Slide 26
- Slide 27
- Slide 28
- Slide 29
- Slide 30
-