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Physics 202Chapter 31Oct 23, 2007
Faraday’s Law
Faraday’s Law
The final step toignite the industrialuse ofelectromagnetism ona large scale.
Light, toasters, cars,TVs, telephones,iPods, industrialproduction, …
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Faraday’s Law “the emf induced in a circuit is directly proportional to
the time rate of change of the magnetic flux throughthe circuit”
ε= induced emf,
= induced potential difference
The induced emf is then: ε = - d/dt (BA cos θ)
In practice often: ΦB = BA cos θ
Faraday’s Law – Example Assume a loop
enclosing an area Alies in a uniformmagnetic field B
The magnetic fluxthrough the loop is ΦB = BA cos θ
The induced emf is ε = - d/dt (BA cos θ)
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Ways of Inducing an emf The magnitude of B can change with time The area enclosed by the loop can change
with time The angle θ between B and the normal to
the loop can change with time Any combination of the above can occur
Lenz’s Law Faraday’s law indicates that the induced
emf and the change in flux have oppositealgebraic signs
Lenz’s law: the induced current in a loopis in the direction that creates amagnetic field that opposes the change inmagnetic flux through the area enclosedby the loop
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Induced emf by changing B-field- a closer look
An increasing B-field generates acurrent.
That means electrons in the wire“see” a (circular) electric field.The E-field is not caused by astatic electric chargedistribution.
That means (in presence of achanging magnetic field):
Faraday’s law can be written ina general form:
!
E " ds# = $d%
B
dt
!
E " ds# $ 0
Examples on the whiteboard
Changing B-field Motional emf
Demonstration: Thesoda can crusher
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Quick Quiz 31.2
The figure shows thetime dependence of auniform fiel B(t) thatpasses through a fixedloop and is orientedperpendicular to theplane of the loop.
Rank the magnitudes ofthe emf generated inthe loop at thefive instants indicated,fromlargest to smallest.
ε(t)
Quick Quiz 31.8
The figure below shows a circular loop of wire beingdropped toward a wire carrying a current to the left. Thedirection of the induced current in the loop of wire is
(a) clockwise
(b) counterclockwise
(c) zero
(d) impossible to determine
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QUICK QUIZ 31.3A square loop of wire is being pulled at a constant velocitythrough a region of uniform magnetic field pointing into thescreen.In the loop of wirea) there will be a current induced,b) charge separation will occur,c) a and b, ord) neither a nor b.
No Flux change!No induced field!No current!No force!
Animation here
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Generators
Electric generatorstake in energy bywork and transfer itout by electricaltransmission
The AC generatorconsists of a loop ofwire rotated by someexternal means in amagnetic field
Applications of Faraday’s Law –Pickup Coil in electric guitar
A permanent magnet isplaced near the string ofthe guitar and causes aportion of the string tobecome magnetized
When the string vibratesat some frequency, themagnetized segmentproduces a changing fluxthrough the coil.
The induced emf is fed toan amplifier.
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Rotating Loop Assume a loop with N
turns, The flux through the loop
at any time t is ΦB = BAcos θ = BA cos ωt
Resulting emf:
DC Generators
The DC (directcurrent) generatorhas essentially thesame components asthe AC generator
The main differenceis that the contactsto the rotating loopare made using asplit ring called acommutator
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Motors
A motor is a generator operating inreverse
Eddy Currents
Circulating currents callededdy currents are induced inbulk pieces of metal movingthrough a magnetic field
The eddy currents are inopposite directions as theplate enters or leaves thefield
Eddy currents are oftenundesirable because theyrepresent a transformation ofmechanical energy intointernal energy
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Maxwell’s Equations,Introduction
Maxwell’s equations are regarded as thebasis of all electrical and magneticphenomena
Maxwell’s equations represent the laws ofelectricity and magnetism that havealready been discussed, but they haveadditional important consequences
Maxwell’s Equations
!
E " dAS
# =q
$0
B " dAS
# = 0
E " ds# = %d&B
dt
B " ds# = µ0I + $
0µ0
d&E
dt
Gauss’s law (electric)
Gauss’s law inMagnetism
Faraday’s Law
Ampere-Maxwell Law
