Page 70 - Electromagnetics Handbook

points:

d
[ˆ n × E − (v · ˆ n)B] dS =− B dV,
S(t) dt V (t)

d
[ˆ n × H + (v · ˆ n)D] dS = J dV + D dV.
S(t) V (t) dt V (t)
We also find that the two Gauss’s law expressions,

D · dS = ρ dV,
S(t) V (t)

B · dS = 0,
S(t)
remain valid.

2.5.3 Large-scale form of the Boffi equations
The Maxwell–Boffi equations can be written in large-scale form using the same ap-
proach as with the Maxwell–Minkowski equations. Integrating (2.120) and (2.121) over
an open surface S and applying Stokes’s theorem, we have

∂B
E · dl =− · dS, (2.161)
S ∂t

∂
0 E

B · dl = µ 0 J + J M + J P + · dS, (2.162)
S ∂t
for fields in the laboratory frame, and
∂B

E · dl =− · dS ,
S ∂t

∂
0 E

B · dl = µ 0 J + J + J + · dS ,
M
P
S ∂t
for fields in a moving frame. We see that Faraday’s law is unmodified by the introduction
of polarization and magnetization, hence our prior discussion of emf for moving contours
remains valid. However, Ampere’s law must be interpreted somewhat differently. The
flux vector B also acts as a force vector, and its circulation is proportional to the out-
flux of total current, consisting of J plus the equivalent magnetization and polarization
currents plus the displacement current in free space, through the surface bounded by the
circulation contour.
The large-scale forms of the auxiliary equations can be found by integrating (2.122)
and (2.123) over a volume region and applying the divergence theorem. This gives

1
E · dS = (ρ + ρ P ) dV,
S
0 V

B · dS = 0,
S
for the laboratory frame fields, and
1

E · dS = (ρ + ρ ) dV ,
P
S
0 V

B · dS = 0,

S
© 2001 by CRC Press LLC