Page 125 - Electromagnetics

2.18 Consider an isotropic but inhomogeneous material, so that

D(r, t) =
(r)E(r, t), B(r, t) = µ(r)H(r, t).

Show that the wave equations for the fields within this material may be written as

2
∂ E ∇
∇µ ∂J
ρ
2
∇ E − µ
+∇ E · − (∇× E) × = µ +∇ ,
∂t 2
µ ∂t
2
∂ H ∇µ ∇
∇
2
∇ H − µ
+∇ H · − (∇× H) × =−∇ × J − J × .
∂t 2 µ
2.19 Consider a homogeneous, isotropic material in which D =
E and B = µH. Using
the definitions of the equivalent sources, show that the wave equations (2.322)–(2.323)
are equivalent to (2.314)–(2.315).
2.20 When we calculate the force on a conductor produced by an incident plane wave,
we often neglect the momentum term

∂
(D × B).
∂t
Compute this term for the plane wave field (2.336) in free space at the surface of the
conductor and compare to the term obtained from the Maxwell stress tensor (2.341).
What is the relative difference in amplitude?
2.21 When a material is only slightly conducting, and thus is very small, we often
neglect the third term in the plane wave solution (2.343). Reproduce the plot of Figure
2.8 with this term omitted and compare. Discuss how the omitted term affects the shape
of the propagating waveform.

2.22 A total charge Q is evenly distributed over a spherical surface. The surface
expands outward at constant velocity so that the radius of the surface is b = vt at time
t. (a) Use Gauss’s law to find E everywhere as a function of time. (b) Show that E may
be found from a potential function

Q
ψ(r, t) = (r − vt)U(r − vt)
4πr
according to (2.361). Here U(t) is the unit step function. (c) Write down the form of
J for the expanding sphere and show that since it may be found from (2.359) it is a
nonradiating source.

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