Chapter 31
Faraday’s Law
Introduction
• This section we will focus on the last of the
fundamental laws of electromagnetism, called
Faraday’s Law of Induction
• Michael Faraday 1791-1867
– Determined Laws of Electrolysis
– Invented electric motor,
generator, and transformer.
Introduction
• In this chapter we will look at the processes in
which a magnetic field (more importantly, a
change in the magnetic field) can induce an
electric current.
31.1 Faraday’s Law of Induction
• An emf and therefore, a current can be
induced in a circuit with the use of a magnet.
• The magnetic field by itself is not capable of
inducing a current.
31.1
• A change in the magnetic field is necessary.
– As the magnet is moved towards the current loop
a positive current is measured.
31.1
– As the magnet is moved away from loop a
negative current is measured.
– Note that this also applies to stationary magnets
and moving coils.
31.1
• Here is the basic setup of actual experiment
conducted by Faraday to confirm this
phenomenon.
31.1
• With the use of insulated wires, the first
circuit and battery is completely isolated from
the second circuit with the ammeter.
– With the 1st circuit open, there is no reading in the
ammeter.
– With the 1st circuit closed, there is no
reading in the ammeter.
31.1
• The instant the switch is open, the ammeter
needle deflects to one side and returns to
zero.
• The instant the switch is closed the ammeter
needle deflects to the opposite side and
returns to zero.
31.1
• So its not the magnetic field that induces the
current, but the change in magnetic field.
• Faraday’s Law of Induction
– The emf induced in a circuit is directly
proportional to the time rate of change of
magnetic flux through the circuit.
d B
 
dt
31.1
• If the circuit is a coil with N number of loops
of the same area, then
d
  N
B
dt
• Assuming a uniform magnetic field the
magnetic flux is equal to BAcosθ so
d
   BA cos  
dt
31.1
• So there are several things that change if
there is going to be an induced current.
– The magnitude of B can change with time.
– The area enclosed by the loop can change with
time.
– The angle , between B and the area vector can
change with time.
– Any combination of the above.
31.1
• Quick quizzes p. 970-971
• Applications of Faraday’s Law
– GFI- induced current in the
coil trips the circuit breaker.
31.1
– Electric Guitar Pickups- the vibrating metal string
induces a current in the coil.
31.1
• Example 31.1, 31.2
31.2 Motional EMFs
• Motional EMF- induced in a conductor moving
through a constant magnetic field.
• Consider a conductor length ℓ, moving
through a constant magnetic field B, with
velocity v.
31.2
• The first thing we notice is that any free
electrons (charge carriers) will feel a magnetic
force as per FB = qv x B
• This will leave one end of the conductor with
extra electrons, and the other with a deficit.
• This creates an electric field within the
conductor which enacts a force on the
electrons opposite of the magnetic force.
31.2
• The forces up and down will balance giving
qE  qvB
E  vB
• The electric field is associated with the
potential difference and the length of the
conductor
V  E  Bv
• This potential difference is maintained as long
as the conductor continues to move with
velocity v through the field.
31.2
• A more interesting example occurs when the
conducting bar is part of a closed circuit.
• We assume zero resistance
in the bar.
• The rest of the circuit has
resistance R.
31.2
• With the magnetic field present, and the
conducting bar free to slide along the
conducting rails, the same potential difference
or EMF is produced, which drives a current
through the circuit.
31.2
• This is another example of Faraday’s law
where the induced current is proportional to
the changing magnetic flux (increasing area).
• Because the area at any time is A = ℓx, the
magnetic flux is given as
 B  Bx
31.2
• From Faraday’s Law, the EMF will be
 
d B
dt
 
d
Bx 
dt
   B
dx
dt
  Bv
31.2
• From this result and Ohm’s law, the induced
current will be

Bv
I 
R
R
• The source of the energy is the work done by
the applied force.
B 2 2v 2  2
P  FAv  IB v 

R
R
31.2
• Quick Quizzes p 975
• Ex 31.4, 31.5
31.3 Lenz’s Law
• Faraday’s Law indicates that the induced emf
and the change in flux have opposite signs.
• This physical effect is known as Lenz’s Law
– The induced current in a loop
is in the direction that creates a
magnetic field that opposes the
change in magnetic flux through
the area enclosed by the loop.
31.3
• We will look at the sliding conductor example
to illustrate.
• In this picture the magnetic
flux is increasing.
• Since the magnetic field is
into the page, the current
induced creates a magnetic
field out of the page.
31.3
• If we switch the direction of travel for the bar,
the flux through the loop is decreasing.
• The current is induced to
oppose that change and
creates additional magnetic
field into the page.
31.3
• We can examine the bar magnet and loop
example again.
31.3
31.3
• Quick Quizzes p. 979
• Conceptual Example 31.6
• Induced Current
– The instant the switch closes
– After a few seconds
– The instant the switch is
opened.
31.4 Induced EMF and Electric Fields
• An E-field within a conductor is responsible for
moving charges through circuit.
• Since Faraday’s law discusses induced
currents, we can claim that the changing
magnetic field creates an E-field within the
conductor.
31.4
• In fact, a changing magnetic field generates an
electric field even without a conducting loop.
• The E-field is however non-conservative unlike
electrostatic fields.
• The work to move a charge
around the loop is given as
  Fd  qE2r 
W q
31.4
• The electric field in the ring is given as
E

2r
• Knowing this and the fact that
 B  BA  r 2 B
• We can apply Faraday’s Law to get
1 d B
r dB
E

2r dt
2 dt
31.4
• So if we have B as a function of time, the
induced current can easily be determined.
• The emf for any closed path can be given as
the line integral of E.ds so Faraday’s Law is
often given in the general form
d B
 E  ds   dt
31.4
• The most important conclusion from this is
the fact that a changing magnetic field,
creates and electric field.
• Quick quiz p 982
• Example 31.8
31.5 Generators and Motors
• Faraday’s Law has a primary application in
Generators and Motors
• AC Generator– Work is done to rotate a loop of wire in a
magnetic field.
– The changing magnetic flux creates an emf that
alternates between positive and negative.
31.5
31.5
• If we look at our rotating loop, the flux
through single turn is given as
 B  BA cos
• And assuming a constant
rotational speed of ω,
  t
• Where θ = 0 at t = 0.
 B  BA cos t
31.5
• If we have more than 1 loop, say N loops, then
Faraday’s Law gives the emf produced as
d B
  N
dt
d
   NAB cos t 
dt
  NAB sin t
31.5
• The maximum emf produced is given as
 max  NAB
• When ωt = 90o and 270o
• Omega is named the angular frequency and is
given as ω = 2πf, where f is the frequency in
Hz.
• Commercial generators in the US operate at f
= 60 Hz.
31.5
• Quick Quiz p. 984
• Example 31.9
• DC Generators
– Operation very similar two AC Generators
– Instead of 2 rings, a DC generator uses one split
ring, called a commutator.
31.5
• Commutator flips the polarity of the brushes
in sync with the rotating loop, ensuring all emf
is of one sign.
• While the emf is always
positive, it pulses with time.
31.5
• Pulsing DC current is not suitable for most
applications, so multiple coil/commutator
combos oriented at different angles are used
simultaneously.
• By superimposing the emf pulses, we get a
very nearly steady value.
31.5
• Motors- Make use of electrical energy to do
work.
• Generator operating in reverse– Current is supplied so a
loop in a magnetic field.
– The torque on the loop
causes rotation which can
be applied to work.
31.5
• The problem is we also have an emf induced
because the magnetic flux changes as the loop
rotates.
• From Lenz’s law this emf opposes the current
running through the loop and is typically
called a “Back emf”
31.5
• When the motor is initially turned on the back
emf is zero.
• As it speeds up the back emf increases.
• If a load is attached to the motor (to do work)
the speed will drop and therefore back emf
will as well.
• This draws higher than normal current from
the voltage source running the motor.
31.5
• If the load jams the motor, and it stops the
motor can quickly burn out, from the
increased current draw.
• Example 31.10
31.6 Eddy Currents
• Eddy Current- A circular
current induced in a bulk
piece of conductor moving
through magnetic field.
31.6
• By Lenz’s law the induced current opposes the
changing flux and therefore creates a
magnetic field on the conductor, that opposes
the source magnetic field.
• Because of this the passing conductor behaves
like an opposing magnetic and the force is
resistive.
31.6
31.6
• The concept is applied to mass transit braking
systems which combine electromagnetic
induction and Eddy currents to steadly slow
subways/trains etc.
• Quick Quiz p. 987
31.7 Maxwell’s Equations
• James Clerk Maxwell developed a list of
equations summarizing the fundamental
nature of electricity and magnetism.
– Gauss’s Law (Electric Fields)
q
 E  dA  
S
o
• The total electric flux through a closed surface is
proportional to the charge contained.
31.7
– Gauss’s Law (Magnetic Fields)
 B  dA  0
S
• The total magnetic flux through a closed surface is zero.
• Magnetic Monopoles have never been observed.
– Faraday’s Law of Induction
 E  ds  
d B
dt
• Electric Fields are created by changing magnetic flux
31.7
– Ampere-Maxwell Law
d E
 E  ds  o I   o o dt
• Magnetic Fields are created by current
• Magnetic Fields are created by changing electric flux.
31.7
• These 4 equations when joined with the
Lorentz Force Law (below) completely
describe all classical electromagnetic
interactions.
F  qE  qv  B
• They are as fundamental to the understanding
of the physical world as Newton’s Laws of
Motion/Universal Gravitation