Page 55 - Electromagnetics Handbook

¯
cD = γ ¯α −1 · (cD) + γ β · H, (2.103)

¯

H =−γ β · (cD) + γ ¯α −1 · H, (2.104)
and

cρ = γ(cρ − β · J), (2.105)

J = ¯α · J − γ βcρ, (2.106)
¯
where ¯α −1 · ¯α = I, and thus ¯α −1 = ¯α − γ ββ.
Maxwell’s equations are covariant under a Lorentz transformation but not under a
Galilean transformation; the laws of mechanics are invariant under a Galilean transfor-
mation but not under a Lorentz transformation. How then should we analyze interactions
between electromagnetic fields and particles or materials? Einstein realized that the laws
of mechanics needed revision to make them Lorentz covariant: in fact, under his theory of
special relativity all physical laws should demonstrate Lorentz covariance. Interestingly,
charge is then Lorentz invariant, whereas mass is not (recall that invariance refers to a
quantity, whereas covariance refers to the form of a natural law). We shall not attempt
to describe all the ramifications of special relativity, but instead refer the reader to any
of the excellent and readable texts on the subject, including those by Bohm [14], Einstein
[62], and Born [18], and to the nice historical account by Miller [130]. However, we shall
examine the importance of Lorentz invariants in electromagnetic theory.
Lorentz invariants. Although the electromagnetic fields are not Lorentz invariant
(e.g., the numerical value of E measured by one observer differs from that measured by
another observer in uniform relative motion), several quantities do give identical values
regardless of the velocity of motion. Most fundamental are the speed of light and the
quantity of electric charge which, unlike mass, is the same in all frames of reference.
Other important Lorentz invariants include E · B, H · D, and the quantities
2
B · B − E · E/c ,
2
H · H − c D · D,
B · H − E · D,
cB · D + E · H/c.

(See Problem 2.3.) To see the importance of these quantities, consider the special case
of fields in empty space. If E·B = 0 in one reference frame, then it is zero in all reference
2
frames. Then if B · B − E · E/c = 0 in any reference frame, the ratio of E to B is
2
always c regardless of the reference frame in which the fields are measured. This is the
characteristic of a plane wave in free space.
2
2
2
If E · B = 0 and c B > E , then we can find a reference frame using the conversion
formulas (2.101)–(2.106) (see Problem 2.5) in which the electric field is zero but the
magnetic field is nonzero. In this case we call the fields purely magnetic in any reference
2
2
2
frame, even if both E and B are nonzero. Similarly, if E · B = 0 and c B < E then
we can find a reference frame in which the magnetic field is zero but the electric field is
nonzero. We call fields of this type purely electric.
The Lorentz force is not Lorentz invariant. Consider a point charge at rest in the
laboratory frame. While we measure only an electric field in the laboratory frame, an
inertial observer measures both electric and magnetic fields. A test charge Q in the

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