Page 54 - Electromagnetics Handbook

Comparison with (2.81)–(2.83) shows that form invariance of Faraday’s law under the
Lorentz transformation requires

E = E x , E = γ(E y − vB z ), E = γ(E z + vB y ),
x y z
and
v v

B = B x , B = γ B y + E z , B = γ B z − E y .

x y 2 z 2
c c
To generalize v to any direction, we simply note that the components of the fields parallel
to the velocity direction are identical in the moving and laboratory frames, while the
components perpendicular to the velocity direction convert according to a simple cross
product rule. After similar analyses with Ampere’s and Gauss’s laws (see Problem 2.2),
we find that

E = E , B = B , D = D , H = H ,

E = γ(E ⊥ + β × cB ⊥ ), (2.87)

⊥

cB = γ(cB ⊥ − β × E ⊥ ), (2.88)
⊥

cD = γ(cD ⊥ + β × H ⊥ ), (2.89)
⊥

H = γ(H ⊥ − β × cD ⊥ ), (2.90)
⊥
and

J = γ(J − ρv), (2.91)

J = J ⊥ , (2.92)
⊥

cρ = γ(cρ − β · J), (2.93)
where the symbols and ⊥ designate the components of the field parallel and perpen-
dicular to v, respectively.
These conversions are self-consistent, and the Lorentz transformation is the transfor-
2
2
mation under which Maxwell’s equations are covariant. If v c , then γ ≈ 1 and to
first order (2.87)–(2.93) reduce to (2.61)–(2.66). If v/c 1, then the first-order fields
reduce to the Galilean fields (2.49)–(2.54).
To convert in the opposite direction, we can swap primed and unprimed fields and
change the sign on v:

E ⊥ = γ(E − β × cB ), (2.94)
⊥ ⊥

cB ⊥ = γ(cB + β × E ), (2.95)
⊥ ⊥

cD ⊥ = γ(cD − β × H ), (2.96)
⊥ ⊥

H ⊥ = γ(H + β × cD ), (2.97)
⊥ ⊥
and

J = γ(J + ρ v), (2.98)

J ⊥ = J , (2.99)
⊥
cρ = γ(cρ + β · J ). (2.100)

The conversion formulas can be written much more succinctly in dyadic notation:
¯

E = γ ¯α −1 · E + γ β · (cB), (2.101)
¯
cB =−γ β · E + γ ¯α −1 · (cB), (2.102)

© 2001 by CRC Press LLC