Page 92 - Electromagnetics Handbook

inclusion of a thermodynamic system in order that energy be conserved. Of course, the
detailed workings of the thermodynamic system are often ignored, indicating that any
thermodynamic “feedback” mechanism is weak. In the waveguide example, for instance,
the heating of the metallic walls does not alter their electromagnetic properties enough
to couple back into an effect on the fields in the walls or in the guide. If such effects were
important, they would have to be included in the conservation theorem via the bound-
ary fields; it is therefore reasonable to associate with these fields a “flow” of energy or
momentum into V . Thus, we wish to develop conservation laws that include not only the
Lorentz force effects within V , but a flow of external effects into V through its boundary
surface.
To understand how external influences may effect the electromagnetic subsystem, we
look to the behavior of the mechanical subsystem as an analogue. In the electromagnetic
system, effects are felt both internally to a region (because of the Lorentz force effect) and
through the system boundary (by the dependence of the internal fields on the boundary
fields). In the mechanical and thermodynamic systems, a region of mass is affected both
internally (through transfer of heat and gravitational forces) and through interactions
occurring across its surface (through transfers of energy and momentum, by pressure
and stress). One beauty of electromagnetic theory is that we can find a mathematical
symmetry between electromagnetic and mechanical effects which parallels the above con-
ceptual symmetry. This makes applying conservation of energy and momentum to the
total system (electromagnetic, thermodynamic, and mechanical) very convenient.

Conservation of momentum and energyin mechanical systems. We begin by
reviewing the interactions of material bodies in a mechanical system. For simplicity we
concentrate on fluids (analogous to charge in space); the extension of these concepts to
solid bodies is straightforward.
Consider a fluid with mass density ρ m . The momentum of a small subvolume of the
fluid is given by ρ m v dV , where v is the velocity of the subvolume. So the momentum
density is ρ m v. Newton’s second law states that a force acting throughout the subvolume
results in a change in its momentum given by
D
(ρ m v dV ) = f dV, (2.272)
Dt
where f is the volume force density and the D/Dt notation shows that we are interested
in the rate of change of the momentum as observed by the moving fluid element (see
§ A.2). Here f could be the weight force, for instance. Addition of the results for all
elements of the fluid body gives
D
ρ m v dV = f dV (2.273)
Dt V V
as the change in momentum for the entire body. If on the other hand the force exerted
on the body is through contact with its surface, the change in momentum is
D
ρ m v dV = t dS (2.274)
Dt V S
where t is the “surface traction.”
We can write the time-rate of change of momentum in a more useful form by applying
the Reynolds transport theorem (A.66):
D ∂
ρ m v dV = (ρ m v) dV + (ρ m v)v · dS. (2.275)
Dt V V ∂t S

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