Ph 632 Problem Set #4 - Department of Physics | Oregon State

Ph 632 Problem Set #4
A. W. Stetz
February 15, 2008
These problems are due Friday, February 22.
1. An ideal circular parallel plate capacitor of radius a and plate separation d a is connected to a current source by axial leads. The current
in the wire is I(t) = I0 cos ωt. (Jackson problem 6.14)
(a) Calculate the electric and magnetic fields between the plates to
second order in powers of the frequency (or wave number), neglecting the effects of fringing fields.
(b) Calculate the volume integrals we and wm that enter in the definition of the reactance X to second order in ω. As we showed in
class
Z
4ω
(wm − we )d3 x
X=
|Ii |2 V
1
1
we = (E · D∗ )
wm = (B · H∗ )
4
4
Show that
Z
Z
|Ii |2 d
|Ii |2 d
ω 2 a2
3
3
we d x = 2 2
wm d x =
1+
ω a
8c2
12c2
Here Ii = −iωQ, where Q is the total charge on one plate.
2. Consider the Dirac expression
Z
A(R) = −g
0
dl0 × ∇(1/|r − r0 |)
−∞
for the vector potential of a magnetic monopole and its associated
string as discussed in class. Suppose as we did that the monopole is
located at the origin and the string along the negative z axis. (Jackson
6.18)
1
(a) Verify my claim that
g ϕ̂
A=
r
1 − cos θ
sin θ
(b) Verify that B = ∇ × A is the Coulomb-like field of a point charge,
except perhaps at θ = π.
(c) With the B determined in part b, evaluate the total magnetic
flux passing through the circular loop of radius R sin θ centered
on the z axis. Consider θ < π/2 and θ > π/2 separately, but
always calculate the upward flux.
H
(d) From A · dl around the loop, determine the total magnetic flux
through the loop. Compare the result with that found in part c.
Show that they are equal for 0 < θ < π/2, but have a constant
difference from π/2 < θ < π. Interpret this difference.