1. An inductor is formed by winding N turns of a perfectly conducting wire into a
circular loop of radius a. The inductor loop is in the x-y plane with its center at the
origin, and connected to a resistor R. In the presence of a magnetic field
B = ẑ (B0 + B1cost) + ŷ B2cost,
where t is time, find
10 1) An expression for the magnetic flux linking a single turn;
10 2) An expression for the induced emf;
5
3) The induced emf, given that N = 10, B0 = 0.5 T, B1 = 0.2 T, B2 = 0.1 T, a = 5 cm,
and = 103 rad/s;
5 4) The induced current in the circuit for R = 1 k.
z
ω
3)
4)
4)
2. Two infinitely long parallel wires (as in the parallel wire transmission line) of radius a
carry currents of equal magnitude I but in opposite directions. The distance between the
axes of the two wires is d.
5
(1) What is the magnetic field H1(x) due to the wire on the left in the figure, at a distance
x from the center of that wire?
10
(2) What is the force between the two wires, assuming d >> a (meaning you can ignore
the finite radii of the wires)? Derive an expression rather than copy it from the book.
Do the two wires attract or repel each other?
10
(3) What is total magnetic field due to the two wires at a distance x from the center of the
left wire?
5
(4) Find the total flux per unit length threading through the area between the two lines.
5
(5) Find the inductance per unit length of the parallel wire transmission line.
2a
(1) By Ampere’s law, 2xH1 = I,
d x
x
Therefore, H1 = I/(2x).
d
(2) Consider the right wire in the field of the left wire, B1(d) =  H1(d) =  I/(2d).
1
1
The two wires, carrying opposite currents, repel each other.
(3)
(4)
(5)
3. The figure below shows a useful way to measure the ac current on a wire without
breaking the circuit or using a current meter. Definitions of dimensions are all shown in
the figure.
From Ampere’s law, you know the magnetic field around the wire is H = I/(2r), where r
is the distance from the wire, whose diameter is negligible. The wire length is much
longer than the dimensions of the iron core, and therefore can be considered infinite. The
iron core has a relative permeability r.
5
(a) Find an expression for the field B(r) inside the iron core.
10 (b) Find the magnetic flux  in the core. Note: Since the field is not uniform, you
need to take an integral from a to b.
10 (c) The coil wound around the core in the figure has N turns. The current carried by
the wire is I = I0cost. Find the expression for the emf.
10 (d) How do you use this setup to measure the current carried by the wire?
coil
(a)
(b)
(c)
(d)