Page 68 - Electromagnetics Handbook

∗
∗
Finally, letting H = H − v × D and J = J − ρv we can write the kinematic form of
Ampere’s law:
d

∗ ∗
H · dl = D · dS + J · dS. (2.154)
dt S S
In a Galilean frame where we use (2.49)–(2.54), we see that (2.154) is identical to
d

H · dl = D · dS + J · dS (2.155)
dt S S
where the primed fields are measured in the frame of the moving contour. This equiv-
alence does not hold when the Lorentz transformation is used to represent the primed
fields.

Alternative form of the large-scale Maxwell equations. We can write Maxwell’s
equations in an alternative large-scale form involving only surface and volume integrals.
This will be useful later for establishing the field jump conditions across a material or
source discontinuity. Again we begin with Maxwell’s equations in point form, but instead
of integrating them over an open surface we integrate over a volume region V moving
with velocity v (Figure 2.3). In the laboratory frame this gives

∂B

(∇× E) dV =− dV,
V V ∂t
∂D

(∇× H) dV = + J dV.
V V ∂t
An application of curl theorem (B.24) then gives

∂B
(ˆ n × E) dS =− dV, (2.156)
S V ∂t
∂D

(ˆ n × H) dS = + J dV. (2.157)
∂t
S V
Similar results are obtained for the fields in the moving frame:
∂B

(ˆ n × E ) dS =− dV ,
S V ∂t

∂D

(ˆ n × H ) dS = + J dV .
S V ∂t
These large-scale forms are an alternative to (2.141)–(2.144). They are also form-
invariant under a Lorentz transformation.
An alternative to the kinematic formulation of (2.153) and (2.154) can be achieved
by applying a kinematic identity for a moving volume region. If V is surrounded by a
surface S that moves with velocity v relative to the laboratory frame, and if a vector field
A is measured in the laboratory frame, then the vector form of the general transport
theorem (A.68) states that
d ∂A
A dV = dV + A(v · ˆ n) dS. (2.158)
dt V V ∂t S

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