Electromagnetic Theory
Fall 2015
Problem Set 11
(due Monday, December 7, 3:00 pm)
1.
Two circular loops of wire are oriented horizontally, one above the other, centered on
a common axis, with their centers separated by a distance z. The upper loop is very
small (compared to z), with radius a, while the lower loop is larger, with radius b.
(a) Suppose current I flows in the big loop. Find the magnetic flux through the
small loop. (The small loop is so small that you may treat the field of the big loop as
uniform.)
(b) Suppose current I flows in the small loop. Find the magnetic flux through the
big loop. (The small loop is so small that you may treat it as an ideal dipole.)
(c) Find the mutual inductances, and confirm that M12 = M21 .
2.
Calculate the energy stored in a long cylindrical solenoid with length `, radius R,
current I, and n turns per unit length, in three different ways: (a)H using the formula
1
1
2
2 LI for the energy stored in an inductor; (b) using the formula 2 (A · I)dl
R in terms
of the current and the vector potential; and (c) using the formula (1/2µ0 ) B 2 dτ in
terms of the magnetic field. You may look up the formulas for L, A, and B in your
textbook and/or class notes.
3.
Solar energy is becoming more affordable and popular, but has the major drawback
that at any given location, the sun isn’t always shining. One way around this problem
is to store energy in batteries that are charged during the day and discharged at night,
but batteries have their own drawbacks. Consider, therefore, the possibility of storing
energy in the magnetic field of a superconducting solenoidal coil. About how large
would such a coil have to be, to store enough energy to power a typical U.S. home for
one night? Would it fit in your basement? Feel free to use any reasonable estimates
for the total energy that needs to be stored and for the maximum magnetic field that
can be attained, but state your estimates clearly and explain how you made them.
Feel free to look up some numbers to test whether your estimates are reasonable.
4.
A fat cylindrical wire, radius a, carries a constant current I, uniformly distributed over
its cross section. A narrow gap in the wire, of width w a, forms a parallel-plate
capacitor, as the surfaces on each side of the gap become charged due to the current
flow. Find the magnetic field in the gap, as a function of the distance s away from the
axis.
5.
Consider the electric and magnetic fields
E(r, t) =
1 q
θ(r − vt)r̂,
4π0 r2
B(r, t) = 0,
where θ(x) is the step function, equal to zero when its argument is negative and 1
when its argument is positive. Show that these fields satisfy all four of Maxwell’s
equations, and determine ρ and J. Describe the physical situation that gives rise to
these fields.
6.
Consider the charging capacitor of Problem 4, above.
(a) Find the electric and magnetic fields in the gap, as functions of s and of the
time t. (Assume that the charge on the surfaces is zero at t = 0.)
(b) Find the energy density uem and the Poynting vector S in the gap. Note especially
the direction of S. Check that the continuity equation for energy is satisfied.
(c) Determine the total energy in the gap as a function of time. Calculate the total
power flowing into the gap, by integrating the Poynting vector over the appropriate
surface. Check that the power input is equal to the rate of increase of energy in the
gap, that is, that Poynting’s theorem is satisfied. (In this case W = 0, because there
is no charge in the gap.)
7.
Find the fields, and the charge and current distributions, corresponding to the potentials
1 qt
V (r, t) = 0,
A(r, t) = −
r̂.
4π0 r2
Then use the gauge function λ = −(1/4π0 )(qt/r) to transform these potentials, and
comment on the result.
You boil it in sawdust: you salt it in glue:
You condense it with locusts and tape:
Still keeping one principal object in view—
To preserve its symmetrical shape.
—Lewis Carroll, The Hunting of the Snark