Potential Difference MAGNA International ReGenX & BiTT Presentation
Electric Potential with Integration Potential Difference in a Variable E-field If E varies, we...
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Transcript of Electric Potential with Integration Potential Difference in a Variable E-field If E varies, we...
Electric Potential with Integration
Potential Difference in a Variable E-field
• If E varies, we integrate the potential difference for a small displacement dl over the path from a to b
b
a
ab dVVV
ddV
lE
lE
©2008 by W.H. Freeman and Company
Potential Difference in a Variable E-field
• Like gravity, the electric field is conservative.
• This means that the potential does not depend on the path taken from a to b.
©2008 by W.H. Freeman and Company
Potential vs. Potential Difference
• Like gravitational potential energy, only differences in electrical potential energy have physical meaning.
• We can choose a convenient reference level to be the zero of potential.
• The potential V of a point is the potential difference between that point and the reference level.
Example: Potential of a point charge
• Calculate the electric potential at a distance r away from a point charge q.
• We choose the reference level to be at infinity.
• We integrate along a radius, so dl = dr.
•
2r
kqE
PP rr
EdrdV lE
drr
kqV
Pr
2
r
kq
r
kqV
P
r
kqV
Quick Review: Potential
• The potential difference between two points is the change in potential energy per unit charge if a charge were moved from one point to the other.
• The potential difference can be found using the electric field.
• The potential of a point change is
r
kqV
0q
UV
b
a
dV lE
Potential and Potential Energy
• We take the zero of potential energy the same as the zero of potential.
• Under those conditions, UE = qV, where q is the charged placed at the position with potential V.
• The formula chart gives the potential energy of charge q2 located a distance r away from q1 as
r
qqU E
21
04
1
Superposition of Potentials
• If more than one charge is present, the potential at a point is the sum of the potentials of each charge.
• The formula on the formula chart is
i i
i
r
qV
04
1
Example- Potential of two charges
•
• In this arrangement, r1 = |x| and r2 = |x-a|
•
2
2
1
1
r
kq
r
kqV
ax
kq
x
kqV
21
©2008 by W.H. Freeman and Company
Calculating V for Continuous Charge Distributions
Integrating over a charge distribution
• If instead of point charges, we have a distribution of charge, we treat each small element of charge as a point charge, and integrating over all the charge elements.
r
dqkV
r
kqV
i i
i
V on the Axis of a Ring
elements. charge allfor constant a is
22
r
azr
22
22
az
kQV
dqaz
kV
22
az
dqk
r
dqkV
©2008 by W.H. Freeman and Company
Charge density
• Linear ()– = Q/L = dQ/dL
• Surface ()– = Q/A = dQ/dA
• Volume ()– = Q/V = dQ/dV
Electric Field
• Spherical symmetry E = 1 q
4o r2
• Continuous charge distributiondE = 1 dq
4o r2
For linear distribution
dE = 1 dq
4o r2
dE = 1 dx
4o r2
E = E = dE
For linear distribution
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
P
cylindrical symmetry in electric field
V near a plane of Charge
• We cannot do as we did for electric field, that is, calculate V for a disk, and then let the size of the disk grow to infinity.– This would yield an
infinite potential.
• We can’t put the zero of potential at infinity, for the same reason.
©2008 by W.H. Freeman and Company
V near a plane of Charge
• Instead, we start with the electric field of a plane of charge, and integrate along a path from the plane to a distance x away.
x
We say the potential at the surface of the sheet is V0.
©2008 by W.H. Freeman and Company
Potential near a plane of charge
.on only depends and
constant, is field electric The
2 0
E
0
V
dxdVx
0 02
lE
x
dxV002
002
V
xV
x
©2008 by W.H. Freeman and Company
Potential near a plane of charge
00 2
x
VV 0
V
00 2
VV
©2008 by W.H. Freeman and Company
Potential near a charged shell
• We consider first the potential at a point outside the shell. (r>R)
©2008 by W.H. Freeman and Company
Potential near a charged shell: Outside
2
2
r
drkQdrEdV
r
kQE
rr
r
drkQ
r
drkQV
22
Rrr
kQV
©2008 by W.H. Freeman and Company
Potential near a charged shell: Outside
Rrr
kQV
• Outside a charged shell, the potential is the same as for a point charge.
©2008 by W.H. Freeman and Company
Potential near a charged shell: Inside
RrR
kQV
• “Inside” the shell, there is no charge.
• The field is zero inside.
• If the field is zero, the potential cannot change, so V is what it is at the surface.
©2008 by W.H. Freeman and Company
Energy Storage
Potential Energy of a System of Charges
• Consider a system of two equal charges, q1 and q2.
• Putting the first charge in place requires no energy.
• Putting the second charge requires q2V, where V is the potential of the charges.
• q2 is ½ the total charge Q, so the energy can also be written
• QVU 21
Potential Energy of a System of Charges
• The potential energy of a system of charges qi is given by
i
iiVqU 21
Computing the Electric Field from the Potential
• The electric field points in the direction of greatest change in potential.
• In the one dimensional case,
dxEV x
dx
dVEx
Computing the Electric Field from the Potential
• In general,
dr
dVEr
Computing the Electric Field from the Potential
• In vector calculus notation
z
VE
y
VE
x
VE zyx
kjiE ˆˆˆz
V
y
V
x
VV