Page 62 - Electromagnetics

In the case of the first-order Lorentz transformation we can set γ ≈ 1 to obtain

E = E + v × B, (2.134)
v × E

B = B − , (2.135)
c 2
v × M

P = P − , (2.136)
c 2
M = M + v × P, (2.137)

J = J − ρv. (2.138)
To convert from the moving frame to the laboratory frame we simply swap primed with
unprimed fields and let v →−v.
As a simple example, consider a linear isotropic medium having

D =
0
E , B = µ 0 µ H ,
r
r
in a moving reference frame. From (117) we have

P =
0
E −
0 E =
0 χ E
r
e

where χ =
− 1 is the electric susceptibility of the moving material. Similarly (2.118)

e r
yields
B B B χ

m
M = − =
µ 0 µ 0 µ µ 0 µ
r r
where χ = µ − 1 is the magnetic susceptibility of the moving material. How are P and

r
m
M related to E and B in the laboratory frame? For simplicity, we consider the first-order
expressions. From (2.136) we have

v × M v × B χ
m
P = P + =
0 χ E + .
e
2
c 2 µ 0 µ c
r
2
Substituting for E and B from (2.134) and (2.135), and using µ 0 c = 1/
0 ,wehave

χ v × E
m
P =
0 χ (E + v × B) +
0 v × B − .
e 2
µ c
r
2
2
Neglecting the last term since it varies as v /c ,weget

χ
m
P =
0 χ E +
0 χ + v × B. (2.139)
e e
µ r
Similarly,
χ m χ m

M = B −
0 χ + v × E. (2.140)
e
µ 0 µ r µ r
2.5 Large-scale form of Maxwell’s equations
We can write Maxwell’s equations in a form that incorporates the spatial variation of
the field in a certain region of space. To do this, we integrate the point form of Maxwell’s
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