= Solution
Dot the <Ampère-Maxwell equation> with $E$, dot <Faraday's law> with $B/\mu_0$, and use the stated <vector> identity. The result is the local balance law
$$
\frac{\partial}{\partial t}
\left[
\frac12\left(\epsilon_0E^2+\frac{B^2}{\mu_0}\right)
\right]
+\nabla\cdot S=-J\cdot E,
\qquad
S=\frac1{\mu_0}E\times B.
$$
Here $S$ is the <Poynting vector>. Integration over $V$ gives the <Poynting theorem>
$$
\frac{dU}{dt}
=-\int_{\partial V}S\cdot n\,dS
-\int_VJ\cdot E\,d^3x.
$$
The field energy decreases through outward electromagnetic energy flux and through work done on charges.
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