PHYS 1110 Lecture 8 Professor Stephen Thornton September 20, 2012.
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Transcript of PHYS 1110 Lecture 8 Professor Stephen Thornton September 20, 2012.
PHYS 1110
Lecture 8
Professor Stephen Thornton
September 20, 2012
If there is a current in If there is a current in
the loop in the direction the loop in the direction
shown, the loop will:shown, the loop will:
A) move upA) move up
B) move downB) move down
C) rotate clockwiseC) rotate clockwise
D) rotate counterclockwiseD) rotate counterclockwise
E) both rotate and moveE) both rotate and move
N S
NS
BB field out of North field out of NorthBB field into South field into South
Reading QuizReading Quiz
Look at the north pole: here the
magnetic field points to the rightright and
the current points out of the pageout of the page.
The right-hand rule says that the force
must point upup. At the south pole, the
same logic leads to a downwarddownward force.
Thus the loop rotates clockwiseclockwise.
N S
F
F
A) move upA) move up
B) move downB) move down
C) rotate clockwiseC) rotate clockwise
D) rotate counterclockwiseD) rotate counterclockwise
E) both rotate and moveE) both rotate and move
If there is a current in If there is a current in
the loop in the direction the loop in the direction
shown, the loop will:shown, the loop will:
Reading QuizReading Quiz
Discuss when Midterm Exam 1 will be. Then determine homework 2 due date.
When would you prefer Exam 1?
A)Thursday, September 27B)Tuesday, October 2C)Thursday, October 4
Helical Motion in a Magnetic Field
Remember
F qv B
What happens if we form a loop with the current carrying wire?
• Do demo with wire loop in magnetic field B.
(galvanometer demo)• We find that the loop rotates in opposite directions depending on the direction of the current!
Magnetic Forces on a Current Loop
0F
0F
Forces cause a torque
F I L B
I I
I
Ftotal = 0
F IhB=
Magnetic Torque on a Current Loop
Top viewWire: width w height h
left right
Torque
/ 2 / 2
sin (if at angle)
r F
IhBw IhBwIhwB IABIAB
r
r
F=IhB
Area = A = hw
F I L B
Wires affected are into screen
Magnetic Force on Current Loops• Consider a rectangular loop in a constant
magnetic field. Can also have N loops.
• Can easily find the force on each side of the loop
• Forces cancel but, depending on orientation, there may be a torque
Loop
We define the magnetic dipole moment of the coil to be .IA NIA
Magnetic Torque on a Current Loop
We showed the torque to be
We can rewrite this in vector form to be
where we have used N loops and the magnetic dipole moment,
sinIAB
B N I A B B
.N I A
Potential Energy of Magnetic Dipole
We can show (but not going to do) that the potential energy of a magnetic dipole moment in a magnetic field is
BU B
Magnetic Torque on a Current Loop
The torque rotates loop until vectors are parallel to
A
and A
.B
0B N I A B
B
Loopof wire
Copyright © 2009 Pearson Education, Inc.
A galvanometer takes advantage of the torque on a current loop to measure current; the spring constant is calibrated so the scale reads in amperes. Remember that all analog ammeters use a galvanometer.
Galvanometer
Copyright © 2009 Pearson Education, Inc.
An electric motor uses the torque on a current loop in a magnetic field to turn magnetic energy into kinetic energy.
Do electric motor demo
B
I
A) leftA) left
B) rightB) right
C) zeroC) zero
D) into the pageD) into the page
E) out of the pageE) out of the page
Conceptual QuizConceptual Quiz
A vertical wire carries a current A vertical wire carries a current
and is in a vertical magnetic field. and is in a vertical magnetic field.
What is the direction of the force What is the direction of the force
on the wire? on the wire?
When the current is parallelparallel to
the magnetic field lines, the force
on the wire is zerozero. B
I
A) leftA) left
B) rightB) right
C) zeroC) zero
D) into the pageD) into the page
E) out of the pageE) out of the page
Conceptual QuizConceptual Quiz
A vertical wire carries a current A vertical wire carries a current
and is in a vertical magnetic field. and is in a vertical magnetic field.
What is the direction of the force What is the direction of the force
on the wire?on the wire?
F I B= ´
Hans Oersted, a Danish physicist, discovered this in 1820 while entertaining students and friends at home. He was preparing a physics lecture.
0
Experimental observation
shows 2
for a long straight wire
IB
r
Now we can imagine combining some of these effects.
• We know that a current carrying wire produces a magnetic field.• We also know that a current carrying wire feels a force in a magnetic field.• If we have two wires, can we use one wire to produce a magnetic field at the position of the second wire? Yes!• If the second wire carries a current,
then it should feel a force!• Do demo – next slide.
We have to look closely at fields and forces to see how the forces occur.
2 2 1F I B
We do this experiment to show that current carrying wires exert forces on each other.
Magnetic field due to current moving through a coil of wire.
Note similarity between B of a bar magnet and B of a coil of wire.
The Solenoid
B=0nI
MRI:
Magnetic Resonance Imaging
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Almost 200 years ago, Michael Faraday looked for evidence that a magnetic field would induce an electric current with this apparatus:
Induced EMF
Do demos about induced currents.
Push and pull magnet in and out of coils of wire to show current production.
Induced Current Produced by a Moving Magnet
v
v
Copyright © 2009 Pearson Education, Inc.
Therefore, a changing magnetic field induces an emf/current.
Faraday’s experiment used a magnetic field that was changing because the current producing it was changing; the previous graphic shows a magnetic field that is changing because the magnet is moving.
Induced EMF
Magnetic Induction
B linked by iron bar.
We conclude that it is the change in magnetic flux that causes induced current.
ind
B B AF =
Faraday’s Discovery and the Law of Induction
There are many ways to change the magnetic flux through a surface:
• Move the magnet
• Turning current on or off in one loop induces current in another
• Move the loop
• Change the shape (and the area) of the loop
The Magnetic Flux Through a Loop
Flux is maximum
Flux is zero. magnetic flux
cosB
B
B A
BA
Look at the mathematics.
This is called Faraday’s Law of Induction after Michael Faraday.
is number of turnsDon't worry about the sign. We willhave a better way to find this later.
Induced emf BNt
N
Do some more demos.
1) Magnet through coil again.2) Flash bulb3) LED coil
Lenz’s Law
The induced current will always be in the direction to oppose the change that produced it.
Induced emf Induced currentÛ
Applying Lenz’s Law to a Magnet Moving Toward and Away From a Current Loop
v v
Induced current
Conceputal QuizIn order to change In order to change the magnetic flux the magnetic flux through the loop, through the loop, what would you what would you have to do?have to do?
A) drop the magnetA) drop the magnet
B) move the magnet upwardsB) move the magnet upwards
C) move the magnet sideways a lotC) move the magnet sideways a lot
D) Only A and BD) Only A and B
E) A, B, and CE) A, B, and C
Moving the magnet in any directionany direction would
change the magnetic field through the
loop and thus the magnetic flux.
Conceptual QuizIn order to change In order to change the magnetic flux the magnetic flux through the loop, through the loop, what would you what would you have to do?have to do?
A) drop the magnetA) drop the magnet
B) move the magnet upwardsB) move the magnet upwards
C) move the magnet sideways a lotC) move the magnet sideways a lot
D) only A and BD) only A and B
E) A, B, and CE) A, B, and C
If a North pole moves If a North pole moves toward the loop from above toward the loop from above the page, in what direction is the page, in what direction is the induced current? the induced current?
A) clockwiseA) clockwise
B) counterclockwiseB) counterclockwise
C) no induced currentC) no induced current
Conceptual Quiz
If a North pole moves toward If a North pole moves toward the loop from above the page, the loop from above the page, in what direction is the in what direction is the induced current? induced current?
A) clockwiseA) clockwise
B) counterclockwiseB) counterclockwise
C) no induced currentC) no induced current
The magnetic field of the moving bar
magnet is pointing into the pageinto the page and
getting largerlarger as the magnet moves
closer to the loop. Thus the induced
magnetic field has to point out of the out of the
pagepage. A counterclockwisecounterclockwise induced
current will give just such an induced
magnetic field.
Conceptual Quiz
Follow-up:Follow-up: What happens if the magnet is stationary but the loop moves? What happens if the magnet is stationary but the loop moves?
Motional emf
What happens when we push rod down?
Determining the Direction of an Induced Current
We exert force to push bar down.
Motional emf
B
B
B A B v t
B v tB v
t te
DF = D = D
DF D= = =
D D
V E
B v E
E B
B vI
R R
ee
e
= =
= =
=
= =
Find force and energy
2 2
2 2 2
mechanical
2 2 2 22
electrical
B vF B B
R
B vP Fv
R
Bv B vP I R R
R
BI
R
R
v
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
A wire loop is being A wire loop is being pulled through a pulled through a uniform magnetic field. uniform magnetic field. What is the direction of What is the direction of the induced current? the induced current?
A) clockwiseA) clockwise
B) counterclockwiseB) counterclockwise
C) no induced currentC) no induced current
Conceputal Quiz
Since the magnetic field is uniform, the
magnetic flux through the loop is not magnetic flux through the loop is not
changingchanging. Thus no current is inducedno current is induced.
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
A wire loop is being A wire loop is being pulled through a pulled through a uniform magnetic field. uniform magnetic field. What is the direction of What is the direction of the induced current? the induced current?
A) clockwiseA) clockwise
B) counterclockwiseB) counterclockwise
C) no induced currentC) no induced current
Conceptual Quiz
Follow-up:Follow-up: What happens if the loop moves out of the page? What happens if the loop moves out of the page?
A conducting rod slides on a
conducting track in a constant
B field directed into the page.
What is the direction of the
induced current?
x x x x x x x x x x x
x x x x x x x x x x x
x x x x x x x x x x x
x x x x x x x x x x x
v
A) clockwiseA) clockwise
B) counterclockwiseB) counterclockwise
C) no induced currentC) no induced current
Conceptual QuizConceptual Quiz
A conducting rod slides on a
conducting track in a constant
B field directed into the page.
What is the direction of the
induced current?
x x x x x x x x x x x
x x x x x x x x x x x
x x x x x x x x x x x
x x x x x x x x x x x
v
The B field points into the pageinto the page. The flux is increasingincreasing since the area is increasing. The induced B field opposes this change and therefore points out of the pageout of the page. Thus, the induced current runs counterclockwise,counterclockwise, according to the right-hand rule.
A) clockwiseA) clockwise
B) counterclockwiseB) counterclockwise
C) no induced currentC) no induced current
Conceptual QuizConceptual Quiz
Follow-up:Follow-up: What direction is the magnetic force on the rod as it moves? What direction is the magnetic force on the rod as it moves?
Copyright © 2009 Pearson Education, Inc.
A generator is the opposite of a motor – it transforms mechanical energy into electrical energy. This is an ac generator:
The axle is rotated by an external force such as falling water or steam. The brushes are in constant electrical contact with the slip rings. See next slide.
Electric Generators
An Electrical Generator
Falling water,steam
Produces AC power
Magnetic flux changes!
Current is induced
A Simple Electric Motor/Generator
Do demo
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If the loop is rotating with constant angular velocity ω, the induced emf is sinusoidal:
For a coil of N loops,
Induced power:
(cos ) sinBA t BA tt
w w we eD=- = =
D
0
0
sin
sin
NBA t
t
NBA
w w
w
w
ee ee
=
=
=
( )222sin
NABP I t
R R
ww
ee= = =
A generator has a coil of A generator has a coil of wire rotating in a wire rotating in a magnetic field. If the magnetic field. If the rotation rate increasesrotation rate increases, , how is the how is the maximum maximum output voltageoutput voltage of the of the generator affected?generator affected?
A) increasesA) increases
B) decreasesB) decreases
C) stays the sameC) stays the same
D) varies sinusoidallyD) varies sinusoidally
Conceptual Quiz
The maximum voltage is the leading
term that multiplies sin(sin(tt)) and is
given by = = NBANBA. Therefore, if
increases increases, then must increase must increase
as well.
)sin( tNBA
A generator has a coil of A generator has a coil of wire rotating in a wire rotating in a magnetic field. If the magnetic field. If the rotation rate increasesrotation rate increases, , how is the how is the maximum maximum output voltageoutput voltage of the of the generator affected?generator affected?
A) increasesA) increases
B) decreasesB) decreases
C) stays the sameC) stays the same
D) varies sinusoidallyD) varies sinusoidally
Conceptual Quiz
Conceptual Quiz:Look at the demonstration of the large electromagnet. Observe what happens (spark) when the switch is opened. What best explains this?A) The battery voltage is leaking through.B) The steady current passing through the magnet.C) Induces a large back current (back emf).D) in this case.V
Answer: C
Nature doesn’t want the magnetic flux to change, so it induces a large current (back emf) to produce a magnetic field. This emf results in the spark across the switch.
Inductance and Inductors• Faraday’s Law: Changing current
in a circuit will induce emf in that circuit as well as others nearby
• Self-Inductance: Circuit induces emf in itself (source of back emf)
• Mutual Inductance: Circuit induces emf in second circuit
Inductance magnetic flux depends on current
is called inductance (actually self inductance here).
B
B
LI
IL
t t
L
The inductance L is a proportionality constant that depends on the geometry of the circuit
Changing Current in an Inductor
Switch open. No current flowing.
Switch closed. Inductor opposes magnetic flux change. Induces current to oppose battery current; current rises more slowly.
Inductor
Magnetic field energyWe know that a battery has to do work to cause current to flow. Similarly an inductor has to do work to cause an induced current to flow from 0 to I in time T. This energy comes from the magnetic field.
2
2
1 1
2 21
2The work done is the energy stored in the inductor
av av
av
I LIL
t T
LIP I V I
T
U P T LI W
WU
21
2U LI
There will be a magnetic flux in Loop 1 due to current I1 flowing in Loop 1 and due to current I2 flowing in Loop 2.
1 1 12 2
2 2 21 1
(1)
Similarly,(2)
B
B
L I M I
L I M I
Now it is clearer why we call L self inductance and M mutual inductance.
For example, two nearby coils
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Mutual InductanceA long thin solenoid of length ℓ and cross-sectional area A contains N1 closely packed turns of wire. Wrapped around it is an insulated coil of N2 turns. All the flux from coil 1 (the solenoid) passes through coil 2. The magnetic flux between the two coils is linked. We call this mutual inductance.
Copyright © 2009 Pearson Education, Inc.
Unit of inductance: the henry, H:
1 H = 1 V·s/A = 1 Ω·s.
A transformer is an example of mutual inductance.
Solenoid Self-Induction
Area A
Only depends on geometry.
0
20 0
20, so
B
B
B nI
NBA nNIA An I
LI L An
Consider an inductorFor a solenoid
20
2 2 2 2 2 20 0
0
2 20
0 0
2
0
1 1 1( )
2 2 2
1 1But , so
2 2
magnetic energy energy density
volume 2
B
B
L n A
U LI n A I n I A
B nI U B A B V
Bu
General energy density
2
0
20
22
00
1 general result
2
1
2
1
2
B
E
B E
Bu
u E
Bu u u E
We can produce an emf by using AC voltage and coils.
Show demo of AC coils and light bulb
Do transformer demo
Transformer equation
P
Primary coil:
Secondary coil:
For good transformer,
If resistance is small,
step-up and step-down transformers
and
PP P
SS S
P PS
S S
SS P
P
P P
S S
Nt
Nt
N
N
NV V
N
V N
V N
Copyright © 2009 Pearson Education, Inc.
This is a step-up transformer – the emf in the secondary coil is larger than the emf in the primary:
Lots of applications for transformers,the bug zapper.
Power distribution
Transformers work only if the current is changing; this is one reason why electricity is transmitted as ac.
120 V120 V
What is the voltage What is the voltage
across the lightbulb? across the lightbulb?
A) 30 VA) 30 V
B) 60 VB) 60 V
C) 120 VC) 120 V
D) 240 VD) 240 V
E) 480 VE) 480 V
Conceptual QuizConceptual Quiz
The first transformerfirst transformer has a 2:1 ratio2:1 ratio
of turns, so the voltage doublesvoltage doubles.
But the second transformersecond transformer has a
1:2 ratio1:2 ratio, so the voltage is halvedvoltage is halved
again. Therefore, the end result is
the same as the original voltagesame as the original voltage.
120 V120 V 240 V240 V 120 V120 V
What is the voltage What is the voltage
across the lightbulb? across the lightbulb?
A) 30 VA) 30 V
B) 60 VB) 60 V
C) 120 VC) 120 V
D) 240 VD) 240 V
E) 480 VE) 480 V
Conceptual QuizConceptual Quiz
A) greater than 6 VA) greater than 6 V
B) 6 VB) 6 V
C) less than 6 VC) less than 6 V
D) zeroD) zero
A 6 V battery is connected to one
side of a transformer. Compared
to the voltage drop across coil A,
the voltage across coil B is:
A B6 V
Conceptual Quiz
The voltage across B is zeroThe voltage across B is zero.
Only a changingchanging magnetic flux
induces an emf. Batteries can
provide only dc currentdc current.
A) greater than 6 VA) greater than 6 V
B) 6 VB) 6 V
C) less than 6 VC) less than 6 V
D) zeroD) zero
A B6 V
Conceptual Quiz
A 6 V battery is connected to one
side of a transformer. Compared
to the voltage drop across coil A,
the voltage across coil B is: