Page 56 - Electromagnetics

laboratory frame experiences the Lorentz force F = QE; in an inertial frame the same

charge experiences F = QE + Qv × B (see Problem 2.6). The conversion formulas show

that F and F are not identical.
We see that both E and B are integral components of the electromagnetic field: the
separation of the field into electric and magnetic components depends on the motion
of the reference frame in which measurements are made. This has obvious implications
when considering static electric and magnetic fields.

Derivation of Maxwell’s equations from Coulomb’s law. Consider a point charge
at rest in the laboratory frame. If the magnetic component of force on this charge arises
naturally through motion of an inertial reference frame, and if this force can be expressed
in terms of Coulomb’s law in the laboratory frame, then perhaps the magnetic field can be
derived directly from Coulomb’s and the Lorentz transformation. Perhaps it is possible
to derive all of Maxwell’s theory with Coulomb’s law and Lorentz invariance as the only
postulates.
Several authors, notably Purcell [152] and Elliott [65], have used this approach. How-
ever, Jackson [91] has pointed out that many additional assumptions are required to
deduce Maxwell’s equations beginning with Coulomb’s law. Feynman [73] is critical of
the approach, pointing out that we must introduce a vector potential which adds to the
scalar potential from electrostatics in order to produce an entity that transforms accord-
ing to the laws of special relativity. In addition, the assumption of Lorentz invariance
seems to involve circular reasoning since the Lorentz transformation was originally in-
troduced to make Maxwell’s equations covariant. But Lucas and Hodgson [117] point
out that the Lorentz transformation can be deduced from other fundamental principles
(such as causality and the isotropy of space), and that the postulate of a vector potential
is reasonable. Schwartz [170] gives a detailed derivation of Maxwell’s equations from
Coulomb’s law, outlining the necessary assumptions.

Transformation of constitutive relations. Minkowski’s interest in the covariance of
Maxwell’s equations was aimed not merely at the relationship between fields in different
moving frames of reference, but at an understanding of the electrodynamics of moving
media. He wished to ascertain the effect of a moving material body on the electromagnetic
fields in some region of space. By proposing the covariance of Maxwell’s equations in
materials as well as in free space, he extended Maxwell’s theory to moving material
bodies.
We have seen in (2.101)–(2.104) that (E, cB) and (cD, H) convert identically under a
Lorentz transformation. Since the most general form of the constitutive relations relate
cD and H to the field pair (E, cB) (see § 2.2.2) as

cD
E
¯
= C ,
H cB
this form of the constitutive relations must be Lorentz covariant. That is, in the reference
frame of a moving material we have

cD
E
= C ,
¯
H cB
¯
¯
and should be able to convert [C ] to [C]. We should be able to find the constitutive
matrix describing the relationships among the fields observed in the laboratory frame.

© 2001 by CRC Press LLC