Page 71 - Electromagnetics Handbook

for the moving frame fields. Here we find the force vector E also acting as a flux vector,
with the outflux of E over a closed surface proportional to the sum of the electric and
polarization charges enclosed by the surface.

To provide the alternative representation, we integrate the point forms over V and use
the curl theorem to obtain
∂B

(ˆ n × E) dS =− dV, (2.163)
S V ∂t

∂
0 E
(ˆ n × B) dS = µ 0 J + J M + J P + dV, (2.164)
S V ∂t
for the laboratory frame fields, and

∂B

(ˆ n × E ) dS =− dV ,
S V ∂t

∂
0 E

(ˆ n × B ) dS = µ 0 J + J + J + dV ,
M P
S V ∂t
for the moving frame fields.
The large-scale forms of the Boffi equations can also be put into kinematic form using
either (2.151) or (2.158). Using (2.151) on (2.161) and (2.162) we have
d

∗
E · dl =− B · dS, (2.165)
(t) dt S(t)
1 d

† †
B · dl = µ 0 J · dS + 2 E · dS, (2.166)
(t) S(t) c dt S(t)
where
∗
E = E + v × B,
1
†
B = B − v × E,
c 2
†
J = J + J M + J P − (ρ + ρ P )v.
†
Here B is equivalent to the first-order Lorentz transformation representation of the field
in the moving frame (2.64). (The dagger † should not be confused with the symbol for
the hermitian operation.) Using (2.158) on (2.163) and (2.164) we have
d

[ˆ n × E − (v · ˆ n)B] dS =− B dV, (2.167)
S(t) dt V (t)
and

1 1 d

ˆ n × B + (v · ˆ n)E dS = µ 0 (J + J M + J P ) dV + E dV.
2
c 2 c dt
S(t) V (t) V (t)
(2.168)
In each case the fields are measured in the laboratory frame, and v is measured with
respect to the laboratory frame and may vary arbitrarily over the surface or contour.

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