Page 94 - Electromagnetics Handbook

equation represents a closed system: charge cannot spontaneously appear and add an
extra term to the right-hand side of (1.11). On the other hand, the change in total
momentum at a point can exceed that given by the momentum flowing out of the point
if there is another “source” (e.g., gravity for an internal point, or pressure on a boundary
point).
To obtain a momentum conservation expression that resembles the continuity equa-
tion, we must consider a “subsystem” with terms that exactly counterbalance the extra
expressions on the right-hand side of (2.279). For a fluid acted on only by external
pressure the sole effect enters through the traction term, and [145]
¯
∇· T m =−∇ p (2.281)
where p is the pressure exerted on the fluid body. Now, using (B.63), we can write
¯
−∇ p =−∇ · T p (2.282)
where

¯
T p = pI ¯
¯
and I is the unit dyad. Finally, using (2.282), (2.281), and (2.280) in (2.279), we obtain
∂
¯ ¯
∇· (T k + T p ) + g k = 0
∂t
and we have an expression for a closed system including all possible effects. Now, note
that we can form the above expression as

∂ ∂
¯ ¯
∇· T k + g k + ∇· T p + g p = 0 (2.283)
∂t ∂t
where g p = 0 since there are no volume effects associated with pressure. This can be
viewed as the sum of two closed subsystems
∂
¯
∇· T k + g k = 0, (2.284)
∂t
∂
¯
∇· T p + g p = 0.
∂t
We now have the desired viewpoint. The conservation formula for the complete closed
system can be viewed as a sum of formulas for open subsystems, each having the form
of a conservation law for a closed system. In case we must include the effects of gravity,
¯
for instance, we need only determine T g and g g such that
∂
¯
∇· T g + g g = 0
∂t
and add this new conservation equation to (2.283). If we can find a conservation ex-
pression of form similar to (2.284) for an “electromagnetic subsystem,” we can include
its effects along with the mechanical effects by merely adding together the conservation
laws. We shall find just such an expression later in this section.
We stated in § 1.3 that there are four fundamental conservation principles. We have
now discussed linear momentum; the principle of angular momentum follows similarly.
Our next goal is to find an expression similar to (2.283) for conservation of energy. We
may expect the conservation of energy expression to obey a similar law of superposition.

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