Page 95 - Electromagnetics Handbook

We begin with the fundamental definition of work: for a particle moving with velocity v
under the influence of a force f k the work is given by f k · v. Dot multiplying (2.272) by v
and replacing f by f k (to represent both volume and surface forces), we get
D
v · (ρ m v) dV = v · f k dV
Dt
or equivalently
D 1
ρ m v · v dV = v · f k dV.
Dt 2
Integration over a volume and application of the Reynolds transport theorem (A.66) then
gives

∂ 1 2 1 2
ρ m v dV + ˆ n · v ρ m v dS = f k · v dV.
V ∂t 2 S 2 V
Hence the sum of the time rate of change in energy internal to the body and the flow
of kinetic energy across the boundary must equal the work done by internal and surface
forces acting on the body. In point form,

∂
∇· S k + W k = f k · v (2.285)
∂t
where
1 2
S k = v ρ m v
2
is the density of the flow of kinetic energy and
1 2
W k = ρ m v
2
is the kinetic energy density. Again, the system is not closed (the right-hand side of
(2.285) is not zero) because the balancing forces are not included. As was done with the
momentum equation, the effect of the work done by the pressure forces can be described
in a closed-system-type equation

∂
∇· S p + W p = 0. (2.286)
∂t
Combining (2.285) and (2.286) we have
∂
∇· (S k + S p ) + (W k + W p ) = 0,
∂t
the energy conservation equation for the closed system.
Conservation in the electromagnetic subsystem. We would now like to achieve
closed-system conservation theorems for the electromagnetic subsystem so that we can
add in the effects of electromagnetism. For the momentum equation, we can proceed
exactly as we did with the mechanical system. We begin with

f em = ρE + J × B.
This force term should appear on one side of the point form of the momentum conserva-
tion equation. The term on the other side must involve the electromagnetic fields, since

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