Page 123 - Electromagnetics

2.2 Consider Ampere’s law and Gauss’s law written in terms of rectangular compo-
nents in the laboratory frame of reference. Assume that an inertial frame moves with
velocity v = ˆ xv with respect to the laboratory frame. Using the Lorentz transformation
given by (2.73)–(2.76), show that

cD = γ(cD ⊥ + β × H ⊥ ),
⊥

H = γ(H ⊥ − β × cD ⊥ ),
⊥

J = γ(J − ρv),

J = J ⊥ ,
⊥

cρ = γ(cρ − β · J),
where “⊥” means perpendicular to the direction of the velocity and “ ” means parallel
to the direction of the velocity.
2.3 Show that the following quantities are invariant under Lorentz transformation:
(a) E · B,
(b) H · D,
2
(c) B · B − E · E/c ,
2
(d) H · H − c D · D,
(e) B · H − E · D,
(f) cB · D + E · H/c.
2
2
2
2.4 Show that if c B > E holds in one reference frame, then it holds in all other
2
2
2
reference frames. Repeat for the inequality c B < E .
2
2
2
2.5 Show that if E·B = 0 and c B > E holds in one reference frame, then a reference
2
2
2
frame may be found such that E = 0. Show that if E · B = 0 and c B < E holds in one
reference frame, then a reference frame may be found such that B = 0.
2.6 A test charge Q at rest in the laboratory frame experiences a force F = QE as
measured by an observer in the laboratory frame. An observer in an inertial frame
measures a force on the charge given by F = QE + Qv × B . Show that F
= F and find

the formula for converting between F and F .
2.7 Consider a material moving with velocity v with respect to the laboratory frame of
reference. When the fields are measured in the moving frame, the material is found to be
isotropic with D =
E and B = µ H . Show that the fields measured in the laboratory

frame are given by (2.107) and (2.108), indicating that the material is bianisotropic when
measured in the laboratory frame.
2
2
2.8 Show that by assuming v /c 1 in (2.61)–(2.64) we may obtain (2.111).
2.9 Derive the following expressions that allow us to convert the value of the magneti-
zation measured in the laboratory frame of reference to the value measured in a moving
frame:
M = γ(M ⊥ + β × cP ⊥ ), M = M .

⊥
© 2001 by CRC Press LLC