Page 61 - Electromagnetics Handbook

merely a mapping from the original vector fields of Minkowski’s form, we still have four
vector fields with which to contend. And with these must also be a mapping of the
constitutive relationships, which now link the fields E, B, P, and M. Rather than argue
the actual physical existence of the equivalent sources, we note that a real benefit of
the new view is that under certain circumstances the equivalent source quantities can be
determined through physical reasoning, hence we can create physical models of P and M
and deduce their links to E and B. We may then find it easier to understand and deduce
the constitutive relationships. However we do not in general consider E and B to be in
any way more “fundamental” than D and H.

Covariance of the Boffi form. Because of the linear relationships (2.117) and (2.118),
covariance of the Maxwell–Minkowski equations carries over to the Maxwell–Boffi equa-
tions. However, the conversion between fields in different moving reference frames will
now involve P and M. Since Faraday’s law is unchanged in the Boffi form, we still have

E = E , (2.124)

B = B , (2.125)

E = γ(E ⊥ + β × cB ⊥ ), (2.126)
⊥

cB = γ(cB ⊥ − β × E ⊥ ). (2.127)
⊥
To see how P and M convert, we note that in the laboratory frame D =
0 E + P and
H = B/µ 0 − M, while in the moving frame D =
0 E + P and H = B /µ 0 − M .Thus

P = D −
0 E = D −
0 E = P

and

M = B /µ 0 − H = B /µ 0 − H = M .

For the perpendicular components

D = γ(D ⊥ + β × H ⊥ /c) =
0 E + P =
0 [γ(E ⊥ + β × cB ⊥ )] + P ;
⊥ ⊥ ⊥ ⊥
substitution of H ⊥ = B ⊥ /µ 0 − M ⊥ then gives
P = γ(D ⊥ −
0 E ⊥ ) − γ
0 β × cB ⊥ + γ β × B ⊥ /(cµ 0 ) − γ β × M ⊥ /c

⊥
or

cP = γ(cP ⊥ − β × M ⊥ ).
⊥
Similarly,

M = γ(M ⊥ + β × cP ⊥ ).
⊥
Hence

E = E , B = B , P = P , M = M , J = J ⊥ , (2.128)

⊥
and

E = γ(E ⊥ + β × cB ⊥ ), (2.129)
⊥

cB = γ(cB ⊥ − β × E ⊥ ), (2.130)
⊥
cP = γ(cP ⊥ − β × M ⊥ ), (2.131)

⊥
M = γ(M ⊥ + β × cP ⊥ ), (2.132)

⊥

J = γ(J − ρv). (2.133)

© 2001 by CRC Press LLC