Download PHYS 520B - Electromagnetic Theory

SPRING 2015
February 2, 2016
PHYS 520B - Electromagnetic Theory
Problem Sheet 1
Deadline February 16
When answering a problem, define any symbols that you introduce, discuss any
assumptions or approximations that you use, and provide appropriate discussions,
explanations, and references.
Q. 1
Consider charging up a circular plate capacitor. A steady current, I, will cause a magnetic
field to circulate the wire carrying the steady current. Now think about what happens inside
the capacitor. Explain why the Maxwell displacement current, ID , is a physical necessity
and show that this requires that between the plates, ID = I.
Q. 2
Jackson 6.1(a) [Hint: you will need delta function property #5 on page 26.]
Q. 3
Suppose magnetic charges did exist, then we could rewrite Maxwell’s equations as follows;
∂B
∇ · E = ρe /ǫ0
∇ × E = −µ0 Jm −
∂t
∂E
∂t
Show that these Maxwell equations are invariant under the duality transformation
∇ · B = µ 0 ρm
E′
cB′
cqe′
′
qm
=
=
=
=
∇ × B = µ 0 Je + µ 0 ǫ0
E cos α + cB sin α,
cB cos α − E sin α,
cqe cos α + qm sin α,
qm cos α − cqe sin α,
where c ≡ √ǫ10 µ0 , and α is an arbitrary (constant) rotation angle in “E/B-space”. Note:charge and current densities transform in the same way as qe and qm .
Q. 4
Consider the quasistatic situation in a conducting medium whereby Ohm’s law relates
the electric field to the current density: J = σE, where σ is the conductivity and the material
is additionally characterized by permeability µ.
(i) Derive the equation for the vector potential A(x, t).
(ii) Hence, show that the Fourier transform of the appropriate Green function is given by
G(k, ω) =
e−ık·x
.
k2
−
ıω
µσ
(iii) Now, upon performing an integration in the complex frequency plane, find the integral
expression for the real space Green function.
′ |2
µσ
(iv) Thus, show that this Green function is, G(x, t; x′ , 0) = Θ(t) 4πt
exp −µσ|x−x
.
4t
Here, Θ is the Heaviside (step) function. [Hint: In going from (iii) to (iv) it will prove
convenient to use a standard trick for Gaussian integrals.]