Physics 9
WS M5 (rev. 1.0)
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M-3. Faraday’s Law
Questions for discussion
1.
In the figure below, there is a non-uniform magnetic field pointing into the page.
a ) If you move the metal loop to the right, will a current be induced in it?
b) If so, will the induced current be clockwise or counter-clockwise?
c) Suppose you want to move the loop to the left at constant speed. Will you have to exert any force to do
this? Why or why not?
d) Suppose instead that you want to move the loop upwards. Will a current be induced in it? Why or why
not?
Physics 9
2.
WS M5 (rev. 1.0)
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The diagram below shows an infinitely long current-carrying wire, with two metal loops nearby.
By means of a current generator (not shown), we cause the current in the wire to increase with time.
a ) Will a current be induced in loop 1? If so, will it be clockwise or counter-clockwise?
b) Will a current be induced in loop 2? If so, will it be clockwise or counter-clockwise?
3. A crude current generator consists of a loop of wire with area A and resistance R. The loop is connected
to a handle, so that someone can cause it to rotate within a uniform magnetic field B 0 that points
upwards.
a ) How does this device generate current?
b) Why will you have to do work in order to turn the handle?
Physics 9
WS M5 (rev. 1.0)
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Problems
1.
A rectangular loop of wire has length L, height
H, and resistance R. The loop balances on
one edge and slides frictionlessly with speed v 0.
It encounters a region of uniform magnetic field
B 0 pointing into the page.
There are five time intervals of importance
here:
a ) For each of the five time intervals, determine
the sense of the induced current (clockwise,
counter-clockwise, or no current).
i . The time before the loop enters the field;
i i . The time while the loop is partly within
field;
i i i . The time while the loop is entirely within
field;
iv. The time while the loop is partly out of
field;
v. The time while the loop is entirely outside
field.
2.
the
b) Taking clockwise currents to be positive, sketch a
graph of the induced current versus time.
the
the
c) Do you think that the final speed of the loop
will be less than v0, equal to v 0, or greater than
v 0? Why?
the
The rectangular metal loop has height H ,
width L, and resistance R. Initially the loop is
situated halfway within a uniform magnetic
field B0. You want to pull the loop out of t h e
field at a constant speed v0.
d) Ignoring the effect alluded to in part (c),
determine the magnitude of the induced current
for each of the five time intervals. 
a ) Explain why you are, in fact, going to have to
do work in order to extract the loop. (Try to give
more than one explanation.)
First method: Forces
b) During the time you are pulling the loop out of
the field with speed v0, what is the magnitude
of the induced current in the loop? Is this
clockwise or counter-clockwise?
c) What force does the magnetic field exert on
each segment of the loop during this time?
d) What is the net magnetic force on the loop? In
which direction does it point?
In this problem, we will calculate in two
different ways the amount of work you will
have to do in order to extract the loop.
(continued, next page)
Physics 9
WS M5 (rev. 1.0)
e) In order to pull the loop to the left with
constant speed v0, what force will you have to
exert?
f)
i)
For how long a time will heat energy be
dissipated at this rate?
j)
How much total heat energy will be dissipated
during the time you pull out the loop?
g) How much work will you have to do in order to
extract the loop?
h ) During the time you are pulling the loop out of
the field with speed v0, what is the rate a t
3.
The metal loops shown below are concentric
circles lying in the same plane.
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which heat is being dissipated by the loop’s
resistance?
Over what distance will you have to exert this
force?
Second method: Energy
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k) How much work will you have to do in order to
extract the loop?
l)
Do your answers for part (g) and part (k) agree?

b) How strong is this magnetic field at the center
point?
c) Assume that the small loop is so small that t h e
magnetic field of the large loop is
approximately uniform over its extent. Under
this assumption, what is the flux of t h e
magnetic field through the small loop?
Now imagine that, starting at t=0, the current
generator causes the current in the large loop to
increase at a steady rate, until it reaches 4 i0 a t
time t=T. After time t=T, the current in t h e
large loop remains steady at 4i0.
d) Sketch the current in the large loop as a
function of time.
The large loop of radius 20a is connected to a
current generator. The small loop of radius a is
not connected to anything.
a ) Suppose that a current i0 is flowing clockwise
around the large loop. Sketch the magnetic
field created by the large loop.
e) During the time 0 < t < T when the current in
the large loop is increasing, will a current be
induced in the small loop? Why or why not? I f
so, then will the induced current be clockwise or
counter-clockwise?
f)
If the resistance of the small loop is R, then
find the magnitude of the current induced in t h e
small loop. 
Physics 9
4.
WS M5 (rev. 1.0)
A rectangular metal loop has height H, length
L, and resistance R. It sits a distance D from a
long straight wire.
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The current in the long straight wire varies in
time according to
i(t) = io (1 –
i(t) = 0
t
)
T
for t < T
for t > T.
a ) What is the magnetic flux, Φ B , through the
loop as a function of time? [Careful: t h e
magnetic field is not uniform over the loop.
What must you do to find the flux?]
b) What is the emf induced around the loop as a
function of time?
c) What is the current induced in the loop?
Which way does this current flow? 