Page 58 - Electromagnetics

which is identical to the expression (2.112) obtained by approximating the exact result
to first order. Similar steps produce (2.111). In a Galilean frame where v/c 1, the

expressions reduce to D =
E and B = µ H, and the isotropy of the fields is preserved.
For a conducting medium having

J = σ E

in a moving reference frame, Cullwick [48] shows that in the laboratory frame
¯

J = σ γ [I − ββ] · E + σ γ cβ × B.
For v c we can set γ ≈ 1 and see that

J = σ (E + v × B)
to first order.

Constitutive relations in deforming or rotating media. The transformations
discussed in the previous paragraphs hold for media in uniform relative motion. When
a material body undergoes deformation or rotation, the concepts of special relativity are
not directly applicable. However, authors such as Pauli [144] and Sommerfeld [185] have
maintained that Minkowski’s theory is approximately valid for deforming or rotating
media if v is taken to be the instantaneous velocity at each point within the body.
The reasoning is that at any instant in time each point within the body has a velocity
v that may be associated with some inertial reference frame (generally different for
each point). Thus the constitutive relations for the material at that point, within some
small time interval taken about the observation time, may be assumed to be those of
a stationary material, and the relations measured by an observer within the laboratory
frame may be computed using the inertial frame for that point. This instantaneous rest-
frame theory is most accurate at small accelerations dv/dt. Van Bladel [201] outlines
its shortcomings. See also Anderson [3] and Mo [132] for detailed discussions of the
electromagnetic properties of material media in accelerating frames of reference.

2.4 The Maxwell–Boffi equations

In any version of Maxwell’s theory, the mediating field is the electromagnetic field
described by four field vectors. In Minkowski’s form of Maxwell’s equations we use E,
D, B, and H. As an alternative consider the electromagnetic field as represented by the
vector fields E, B, P, and M, and described by
∂B
∇× E =− , (2.113)
∂t
∂
∇× (B/µ 0 − M) = J + (
0 E + P), (2.114)
∂t
∇· (
0 E + P) = ρ, (2.115)
∇· B = 0. (2.116)

These Maxwell–Boffi equations are named after L. Boffi, who formalized them for moving
media [13]. The quantity P is the polarization vector, and M is the magnetization vector.

© 2001 by CRC Press LLC