Page 52 - Electromagnetics

These can also be written using dyadic notation as
¯
¯

E = I · E + β · (cB), (2.67)
¯
¯
cB =−β · E + I · (cB), (2.68)

and
¯
¯

cD = I · (cD) + β · H, (2.69)
¯
¯

H =−β · (cD) + I · H, (2.70)
where
 
0 −β z β y
¯
[β] =  β z 0 −β x 
0
−β y β x
with β = v/c. This set of equations is self-consistent among Maxwell’s equations. How-
ever, the equations are not consistent with the assumption of a Galilean transformation
of the coordinates, and thus Maxwell’s equations are not covariant under a Galilean
transformation. Maxwell’s equations are only covariant under a Lorentz transforma-
tion as described in the next section. Expressions (2.61)–(2.64) turn out to be accurate
to order v/c, hence are the results of a first-order Lorentz transformation. Only when
v is an appreciable fraction of c do the field conversions resulting from the first-order
Lorentz transformation differ markedly from those resulting from a Galilean transforma-
tion; those resulting from the true Lorentz transformation require even higher velocities
to differ markedly from the first-order expressions. Engineering accuracy is often accom-
plished using the Galilean transformation. This pragmatic observation leads to quite a
bit of confusion when considering the large-scale forms of Maxwell’s equations, as we
shall soon see.

2.3.2 Field conversions under Lorentz transformation

To find the proper transformation under which Maxwell’s equations are covariant,
we must discard our notion that time progresses the same in the primed and the un-
primed frames. The proper transformation of coordinates that guarantees covariance of
Maxwell’s equations is the Lorentz transformation

ct = γ ct − γ β · r, (2.71)

r = ¯α · r − γ βct, (2.72)
where
1 ββ
¯
, ¯ α = I + (γ − 1) , β =|β|.
γ = 2 2
1 − β β
This is obviously more complicated than the Galilean transformation; only as β → 0 are
the Lorentz and Galilean transformations equivalent.
Not surprisingly, field conversions between inertial reference frames are more com-
plicated with the Lorentz transformation than with the Galilean transformation. For
simplicity we assume that the velocity of the moving frame has only an x-component:
v = ˆ xv. Later we can generalize this to any direction. Equations (2.71) and (2.72)
become
x = x + (γ − 1)x − γvt, (2.73)

© 2001 by CRC Press LLC