Page 51 - Electromagnetics Handbook

Equations (2.46), (2.47), and (2.48) show that the forms of Maxwell’s equations in the
inertial and laboratory frames are identical provided that

E = E + v × B, (2.49)

D = D, (2.50)

H = H − v × D, (2.51)

B = B, (2.52)

J = J − ρv, (2.53)

ρ = ρ. (2.54)
That is, (2.49)–(2.54) result in form invariance of Faraday’s law, Ampere’s law, and the
continuity equation under a Galilean transformation. These equations express the fields
measured by a moving observer in terms of those measured in the laboratory frame. To
convert the opposite way, we need only use the principle of relativity. Neither observer
can tell whether he or she is stationary — only that the other observer is moving relative
to him or her. To obtain the fields in the laboratory frame we simply change the sign on
v and swap primed with unprimed fields in (2.49)–(2.54):

E = E − v × B , (2.55)

D = D , (2.56)
H = H + v × D , (2.57)

B = B , (2.58)

J = J + ρ v, (2.59)

ρ = ρ . (2.60)
According to (2.53), a moving observer interprets charge stationary in the laboratory
frame as an additional current moving opposite the direction of his or her motion. This
seems reasonable. However, while E depends on both E and B , the field B is unchanged

under the transformation. Why should B have this special status? In fact, we may
uncover an inconsistency among the transformations by considering free space where
(2.22) and (2.23) hold: in this case (2.49) gives

D /
0 = D/
0 + v × µ 0 H
or

D = D + v × H/c 2
rather than (2.50). Similarly, from (2.51) we get

B = B − v × E/c 2
instead of (2.52). Using these, the set of transformations becomes

E = E + v × B, (2.61)
2
D = D + v × H/c , (2.62)

H = H − v × D, (2.63)

2
B = B − v × E/c , (2.64)

J = J − ρv, (2.65)

ρ = ρ. (2.66)

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