Page 97 - Electromagnetics Handbook

Equation (2.290) is the point form of the electromagnetic conservation of momentum
theorem. It is mathematically identical in form to the mechanical theorem (2.279).
Integration over a volume gives the large-scale form

¯ ∂g em
T em · dS + dV =− f em dV. (2.291)
S V ∂t V
If we interpret this as we interpreted the conservation theorems from mechanics, the first
term on the left-hand side represents the flow of electromagnetic momentum across the
boundary of V , while the second term represents the change in momentum within V . The
sum of these two quantities is exactly compensated by the total Lorentz force acting on
the charges within V . Thus we identify g em as the transport density of electromagnetic
momentum.
Because (2.290) is not zero on the right-hand side, it does not represent a closed system.
If the Lorentz force is the only force acting on the charges within V , then the mechanical
reaction to the Lorentz force should be described by Newton’s third law. Thus we have
the kinematic momentum conservation formula
¯ ∂g k
∇· T k + = f k =−f em .
∂t
Subtracting this expression from (2.290) we obtain
∂
¯ ¯
∇· (T em − T k ) + (g em − g k ) = 0, (2.292)
∂t
which describes momentum conservation for the closed system.
It is also possible to derive a conservation theorem for electromagnetic energy that
resembles the corresponding theorem for mechanical energy. Earlier we noted that v · f
represents the volume density of work produced by moving an object at velocity v under
the action of a force f. For the electromagnetic subsystem the work is produced by
charges moving against the Lorentz force. So the volume density of work delivered to
the currents is

w em = v · f em = v · (ρE + J × B) = (ρv) · E + ρv · (v × B). (2.293)

Using (B.6) on the second term in (2.293) we get
w em = (ρv) · E + ρB · (v × v).

The second term vanishes by definition of the cross product. This is the familiar property
that the magnetic field does no work on moving charge. Hence
w em = J · E. (2.294)

This important relation says that charge moving in an electric field experiences a force
which results in energy transfer to (or from) the charge. We wish to write this energy
transfer in terms of an energy flux vector, as we did with the mechanical subsystem.
As with our derivation of the conservation of electromagnetic momentum, we wish to
relate the energy transfer to the electromagnetic fields. Substitution of J from (2.2) into
(2.294) gives

∂D
w em = (∇× H) · E − · E,
∂t

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