Page 69 - Electromagnetics Handbook

Applying this to (2.156) and (2.157) we have

d

[ˆ n × E − (v · ˆ n)B] dS =− B dV, (2.159)
S dt V

d
[ˆ n × H + (v · ˆ n)D] dS = J dV + D dV. (2.160)
S V dt V
We can also apply (2.158) to the large-scale form of the continuity equation (2.10) and
obtain the expression for a volume region moving with velocity v:

d
(J − ρv) · dS =− ρ dV.
S dt V

2.5.2 Moving, deforming surfaces
Because (2.151) holds for arbitrarily moving surfaces, the kinematic versions (2.153)
and (2.154) hold when v is interpreted as an instantaneous velocity. However, if the
surface and contour lie within a material body that moves relative to the laboratory
frame, the constitutive equations relating E, D, B, H, and J in the laboratory frame
differ from those relating the fields in the stationary frame of the body (if the body is
not accelerating), and thus the concepts of § 2.3.2 must be employed. This is important
when boundary conditions at a moving surface are needed. Particular care must be taken
when the body accelerates, since the constitutive relations are then only approximate.
The representation (2.145)–(2.146) is also generally valid, provided we define the
primed fields as those converted from laboratory fields using the Lorentz transforma-
tion with instantaneous velocity v. Here we should use a different inertial frame for each
point in the integration, and align the frame with the velocity vector v at the instant
t. We certainly may do this since we can choose to integrate any function we wish.
However, this representation may not find wide application.
We thus choose the following expressions, valid for arbitrarily moving surfaces con-
taining only regular points, as our general forms of the large-scale Maxwell equations:

d d (t)

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

∗ ∗
H · dl = D · dS + J · dS,
(t) dt S(t) S(t)
where
E = E + v × B,
∗
∗
H = H − v × D,
∗
J = J − ρv,
and where all fields are taken to be measured in the laboratory frame with v the in-
stantaneous velocity of points on the surface and contour relative to that frame. The
constitutive parameters must be considered carefully if the contours and surfaces lie in
a moving material medium.
Kinematic identity (2.158) is also valid for arbitrarily moving surfaces. Thus we have
the following, valid for arbitrarily moving surfaces and volumes containing only regular

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