Page 65 - Electromagnetics Handbook

Figure 2.3: Non-deforming volume region having velocity v relative to laboratory (un-
primed) coordinate system.

is the large-scale magnetic field Gauss’s law. It states that the total flux of B passing
through a closed surface is zero, since there are no magnetic charges contained within
(i.e., magnetic charge does not exist).
Since charge is an invariant quantity, the large-scale forms of the auxiliary equations
take the same form in a moving reference frame:

D · dS = ρ dV = Q(t) (2.149)
S V
and

B · dS = 0. (2.150)
S
The large-scale forms of the auxiliary equations may be derived from the large-scale
forms of Faraday’s and Ampere’s laws. To obtain Gauss’s law, we let the open surface

in Ampere’s law become a closed surface. Then H · dl vanishes, and application of
the large-scale form of the continuity equation (1.10) produces (2.147). The magnetic
Gauss’s law (2.148) is found from Faraday’s law (2.141) by a similar transition from an
open surface to a closed surface.
The values obtained from the expressions (2.141)–(2.142) will not match those ob-
tained from (2.143)–(2.144), and we can use the Lorentz transformation field conversions
to study how they differ. That is, we can write either side of the laboratory equations in
terms of the moving reference frame fields, or vice versa. For most engineering applica-
tions where v/c 1 this is not done via the Lorentz transformation field relations, but
rather via the Galilean approximations to these relations (see Tai [194] for details on us-
ing the Lorentz transformation field relations). We consider the most common situation
in the next section.

Kinematic form of the large-scale Maxwell equations. Confusion can result from
the fact that the large-scale forms of Maxwell’s equations can be written in a number of

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