Faraday’s Law problems
Problem 1 Starting with the differential form of Faraday’s law,
choose an open surface, take a surface integral of both sides, and
apply Stokes’ theorem to obtain the integral form of Faraday’s
law.
Problem 2 Consider a uniform magnetic field B = (B0 + γt)k̂.
A square wire with side length L lies in the xy plane.
(a) What are the units of the constant γ?
(b) Find the magnitude and direction of the emf around the
loop.
Problem 3 Consider a uniform magnetic field B = (B0 +γt)k̂. A
circular wire with radius r0 and resistance R lies in the xy plane.
Find the magnitude and direction of the current in the loop.
Problem 4 Consider a uniform magnetic field B = B0 k̂. A
circular wire loop is made from a flexible metallic wire. The
radius r = r0 + a cos ωt of the loop oscillates in time. The loop
lies in the xy plane and has a constant resistance R.
(a) What are the units of the constants r0 , a, ω, R, and t?
(b) Find the magnitude and direction of the current in the loop.
Problem 5 Consider a long straight wire lying on the y axis
carrying current I in the positive y direction. A circular loop
with radius r0 lies in the xy plane with its center on the positive
x axis. The circular loop does not overlap the straight wire. If the
circular loop is pulled so that its center moves along the x axis,
in the positive x direction, will a current be induced in the loop?
Why or why not? If a current is induced, in which way will it
flow? How do you know? Draw a picture of the situation.
Problem 6 Consider a circular loop in the xy plane carrying
current I counterclockwise. The center of this loop is at the
origin. A second circular loop with the same radius lies in the
xy plane with its center on the positive x axis. The loops do not
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overlap. If the second circular loop is pulled so that its center
moves along the x axis, in the positive x direction, will a current
be induced in the loop? Why or why not? If a current is induced,
in which way will it flow? How do you know? Draw a picture of
the situation.
Problem 7 Consider the following simplified model of an electric
generator. We have a uniform magnetic field B = B0 ĵ. We have
a single loop of wire that rotates in this magnetic field. You can
make it a circular loop or a square loop, I don’t care. The loop
has area A. At t = 0, the loop lies in the xz plane. At t = T /4,
the loop lies in the yz plane. At t = T /2, the loop lies in the
xz plane again (having made a 180◦ rotation since t = 0). By
t = T , the loop is back where it started, having made a 360◦
rotation since t = 0. The rotation is occuring about the z axis,
with period T .
(a) Give expressions for the magnetic flux through the loop at
times t = 0, t = T /4, t = T /2, t = 3T /4, and t = T .
(b) Give an expression for the magnetic flux through the loop
at any time t.
(c) Give an expression for the emf generated in the loop at time
t.
(d) Give an expression for the maximum emf generated in the
loop.
Problem 8 Consider a square loop with side length L. At t = 0,
the loop is lying in the xz plane with the origin at its center.
The loop rotates about the z axis with period T . At t = T /4,
for example, the loop lies in the yz plane. There is a uniform
magnetic field B = B0 î
(a) Write an expression for the magnetic flux through the loop
at time t.
(b) Find the emf around the loop as a function of time.
Problem 9 A circular loop with radius R rotates with a period
T about an axis along the diameter of the loop. Suppose that this
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diameter is parallel to the z-axis. There is a uniform magnetic
field B = B x̂ in the vicinity of the loop.
(a) Give an expression for the maximum flux through the loop.
(b) Give an expression or make a careful graph (axes labeled,
etc.) showing the flux through the loop as a function of
time.
(c) Give an expression or make a careful graph showing the emf
around the loop as a function of time.
(d) Give an expression for the maximum emf around the loop.
Problem 10 Suppose we have a uniform magnetic field that is
increasing linearly in time.
B(r, t) = αtk̂
Find an electric field that will satisfy Faraday’s law.
Problem 11 Consider a uniform magnetic field
B = (B0 + B1 cos ωt)k̂.
A square wire with side length L lies in the xy plane.
(a) What are the units of the constants B0 , B1 , and ω?
(b) Find the emf around the loop as a function of time. What
does a positive emf mean? What does a negative emf mean?
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