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Chapter 22: Electromagnetic Induction
- Induced emf and current
- Magnetic flux
- Faraday’s and Lenz’s laws
- Electric generators, back emf
- Omit 22.8, 22.9, (inductance and transformers)
Monday, February 19 , 2007 1
Induced emf
When the magnet moves relative
to coil, a current is induced in the
coil.
Reversing the magnet N and S
poles reverses the deflection.
Moving the coil to the magnet
produces the same deflection as
moving the magnet to the coil –
only the relative motion of coil
and magnet matters.
zero
! v
++
I
I
I
I
Charges inside the moving rod experience a force
due to the magnetic field...
Conductor
The moving conductor acts as a generator.
Electromagnetic Induction
The basis of electromagnetic induction.
Monday, February 19 , 2007 3
B
A charge! q inside the wire moves with the coil relative to the magnetic
field. A component of field, B
"
, is perpendicular to the velocity of the coil.
B
⊥
! F
x
F
I
B
⊥
Moving coil relative to magnet
! v
Motion of coil
toward the
magnet
The magnetic forces induce a current to flow around the coil.
Induced emf
The emf induced between the ends of a conductor that is moving in
a magnetic field is:
V = vLB
The induced emf is the same whether the coil moves or the
magnet moves, only the relative motion matters.
( V = vLB sin! when the angle between
B and! v is !)
Monday, February 19 , 2007 7
Prob. 22.2/4: “Tethered Satellite Experiment”. A 20,000 m length of
wire is trailed behind the shuttle while in orbit around the earth. The
orbital speed of the shuttle is 7600 m/s.
If the earth’s magnetic field at the position of the shuttle is 5.1! 10
and the wire moves perpendicular to the field, what is the induced emf
between the ends of the wire?
V = vLB = 7600! 20,000! 5.1! 10
Negative at the top.
B
! v
Wire
Prob. 22.4/2: The drawing shows a type of blood flow meter. Blood is
conductive enough that it can be treated as a moving conductor. When it
flows perpendicular to a magnetic field, electrodes can be used to measure
the small voltage that develops across the vessel.
Suppose the speed of the blood is 0.3 m/s, the diameter of the vessel is 5.
mm and B = 0.6 T. What is the magnitude of the voltage that is measured?
! v
L
B
Blood – a moving
conductor
Monday, February 19 , 2007 9
Prob. 22.5: Each rod of length L = 1.3 m moves at speed v = 2.7 m/s in a
magnetic field, B = 0.45 T. Find the motional emf for each.
The moving rod now generates its own emf that opposes the emf of the
battery (a “back emf”). The current therefore decreases. The rod
continues to accelerate until the current is reduced to zero (assuming no
friction).
v
V = vLB I
V 0 F = ILB
Speed constant when vLB = V 0
Monday, February 19 , 2007 13
V 0 V = vLB
I
I
Induced emf – equivalent circuit
R
Resistance of rails and bar
I = 0 when V = V 0
60 W bulb, R = 240 "
Motional emf between ends of sliding rod, V = vLB
Power dissipated, W = VI = V
2 /R = 60 W, so ( vLB )
2 /R = 60 W
B = 0.4 T L = 0.6 m
Therefore, v =
60 R
LB
60 × ( 240 !)
( 0. 6 m) × ( 0. 4 T)
= 500 m/s
In 0.5 s, the rod slides 250 m!
How long do the
rails have to be to
light the 60 W bulb
for 0.5 s?
F
m
F
applied
Monday, February 19 , 2007 15
B = 0.4 T L = 0.6 m
Work done to light the lamp
There is an induced
current I in the bar
when the bar is moving
in the magnetic field.
Magnetic force on the
bar, F
m
= ILB, opposes
the motion of the bar.
F
applied
In 1 s, work done by the applied force in opposing F m
is W = F m
v
W = F
m
v = ( ILB) v = I ( LBv ) = I V = 60 W
That is, the power to light the bulb is supplied by doing work against the
magnetic force.
F
m
60 W lamp
The emf induced between the ends of the falling
rod is:
V = vLB
No current is flowing, so there is no magnetic
force on the rod.
The resistor R completes the circuit, so that
current flows and there is now a magnetic force
resisting the gravitational force that
accelerates the rod downwards.
The rod stops accelerating when the magnetic
force is equal to the weight of the rod.
Motional emf
L
L
! v
! v
Monday, February 19 , 2007 19
Prob. 22.9: A conducting rod 1.3 m long slides down between two
frictionless vertical copper tracks at a constant speed of 4 m/s
perpendicular to a 0.5 T magnetic field.
a) What is the mass of the rod?
b) Find the change in gravitational PE in 0.2 s.
c) Find the electrical energy dissipated in the resistor in 0.2 s.
R = 0.75 "
L = 1.3 m
Induced emf
Changing the area of the loop also
induces an emf.
An emf is induced in the coil
whenever the number of field lines
passing through the coil changes.
The number of field lines is a
measure of “ magnetic flux ”.
! There is an induced emf
whenever there is a change of
magnetic flux passing through the
coil.
Monday, February 19 , 2007 21
Induced emf
The emf induced between the ends of the moving rod is: V = vLB
Between time t
0
and t, the rod moves a distance x – x
0
= v ( t – t
0
), so
V = vLB = B
( x − x 0
) L
( t − t 0
= B
A − A
0
t − t 0
( BA ) − ( BA )
0
t − t 0
A
0
= x
0
L
A = xL
With B perpendicular to the loop, ( BA ) is the “ magnetic flux ” passing
through the loop. The induced emf is equal to the rate of change of
magnetic flux passing through the loop – Faraday’s Law.
!( BA )
! t
L
A 0 A
L
B = 0.4 T L = 0.6 m
F
applied
F
m
Induced emf and rate of change of flux
As the rod is moved to the right, the area of the closed loop increases and
the magnetic flux passing through the loop increases. There is increasing
magnetic flux passing through the loop and pointing into the page.
R
BI
! B I
! B I
The induced current, I , produces a magnetic field, B I
, pointing out of the
page, that opposes the change in magnetic flux. This is Lenz’s law.
Faraday’s Law:
V =
! t
= IR
= B
! A
! t
= BLv
Monday, February 19 , 2007 25
Faraday’s Law
The induced emf is equal to
the rate of change of
magnetic flux.
The direction of the induced current is such that
the magnetic field produced by the current
opposes the change in magnetic flux that
generated the current.
Lenz’s Law
Magnetic flux:! = BA cos "
Prob. 22.C1 1 : Use Lenz’s law to verify
that the induced current is in the direction
in the diagram.
- The flux through the loop is
into the page and is decreasing as
the area of the loop decreases.
- The induced current produces a
magnetic field that opposes the
decreasing flux.
! The magnetic field produced
by the induced current must
point into the page.
! The current flows clockwise
in the loop
Monday, February 19 , 2007 27
Conducting ring falling through a magnetic field
The magnetic flux passing through the ring
is constant (zero), so there is no induced
emf or current.
! B I
I
I = 0
Fm
A magnetic force is generated that opposes the motion of the ring.
The magnetic flux passing through the
ring is increasing and is directed into
the page.
The induced current produces a
magnetic field, BI, that opposes the
increase of flux.
Faraday’s Law
The induced emf is equal to
the rate of change of
magnetic flux.
The direction of the induced current is such that
the magnetic field produced by the current
opposes the change in magnetic flux that
generated the current.
Lenz’s Law
Magnetic flux:! = BA cos "
Monday, February 19 , 2007 31
Prob. 22.33: A circular loop of wire rests on a table. A long, straight wire
lies on this loop over its centre.
The current I in the straight wire is increasing. In what direction is the
induced current, if any, in the loop?
- What is the total magnetic flux through the loop?
- Does it change?
B
A wire is bent into a circular loop as shown. The radius of the circle is 2
cm. A constant magnetic field B = 0.55 T is directed perpendicular to the
plane of the loop. Someone grabs the ends of the wire and pulls it taut, so
the radius shrinks to zero in 0.25 s.
Find the magnitude of the average induced emf between the ends of the
wire.
B
I
Monday, February 19 , 2007 33
Prob. 22.70/32: Indicate the direction of the electric field between the
plates of the capacitor if the magnetic field is decreasing in time.
E
I
I
I
BI
BI
BI
Induced magnetic field
Guitar pickup
- The strings are magnetizable
- A permanent magnet magnetizes them
- The vibration of a string changes the magnetic flux through a coil
close to it at the frequency of vibration of the string
- An emf is induced in the coil at that frequency
Monday, February 19 , 2007 37
Playback of tape recording
- Recorded tape is magnetized in N-S patches
- The tape passes by the playback head which channels and concentrates
the magnetic field through an iron core
- The changing magnetic flux induces an emf in the coil
Moving coil microphone
- Sound waves cause the diaphragm of the microphone to move in/out
- A coil moves with the diaphragm relative to a permanent magnet,
causing the magnetic flux through the coil to change in step with the
pressure variations of the sound wave
- An emf is set up in the coil at the frequency of the sound wave
Monday, February 19 , 2007 39
Ground fault detector
If the return current (green) is equal to the supply current (red), the magnetic
fluxes around the iron ring are equal and opposite and cancel.
If the currents differ, the fluxes do not
cancel and there is a net flux varying at
60 Hz, which induces a current in the
sensing coil.
