Page 124 - Electromagnetics

2.10 Beginning with the expressions (2.61)–(2.64) for the field conversions under a
first-order Lorentz transformation, show that

v × M

P = P − , M = M + v × P.
c 2
2.11 Consider a simple isotropic material moving through space with velocity v relative
to the laboratory frame. The relative permittivity and permeability of the material

measured in the moving frame are
and µ , respectively. Show that the magnetization
r
r
as measured in the laboratory frame is related to the laboratory frame electric field and
magnetic flux density as

χ m χ m

M = B −
0 χ + v × E
e
µ 0 µ µ
r r
when a first-order Lorentz transformation is used. Here χ =
− 1 and χ = µ − 1.

e r m r
2.12 Consider a simple isotropic material moving through space with velocity v relative
to the laboratory frame. The relative permittivity and permeability of the material

measured in the moving frame are
and µ , respectively. Derive the formulas for the

r r
magnetization and polarization in the laboratory frame in terms of E and B measured in
the laboratory frame by using the Lorentz transformations (2.128) and (2.129)–(2.132).
Show that these expressions reduce to (2.139) and (2.140) under the assumption of a
2
2
first-order Lorentz transformation (v /c 1).
2.13 Derive the kinematic form of the large-scale Maxwell–Boffi equations (2.165) and
(2.166). Derive the alternative form of the large-scale Maxwell–Boffi equations (2.167)
and (2.168).
2.14 Modify the kinematic form of the Maxwell–Boffi equations (2.165)–(2.166) to
account for the presence of magnetic sources. Repeat for the alternative forms (2.167)–
(2.168).
2.15 Consider a thin magnetic source distribution concentrated near a surface S. The
magnetic charge and current densities are given by
ρ m (r, x, t) = ρ ms (r, t) f (x, ), J m (r, x, t) = J ms (r, t) f (x, ),
where f (x, ) satisfies
∞

f (x, ) dx = 1.
−∞
Let → 0 and derive the boundary conditions on (E, D, B, H) across S.
2.16 Beginning with the kinematic forms of Maxwell’s equations (2.177)–(2.178), de-
rive the boundary conditions for a moving surface

ˆ n 12 × (H 1 − H 2 ) + (ˆ n 12 · v)(D 1 − D 2 ) = J s ,
ˆ n 12 × (E 1 − E 2 ) − (ˆ n 12 · v)(B 1 − B 2 ) =−J ms .
2.17 Beginning with Maxwell’s equations and the constitutive relationships for a bian-
isotropic medium (2.19)–(2.20), derive the wave equation for H (2.313). Specialize the
result for the case of an anisotropic medium.

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