= Solution
Dot the <Ampère-Maxwell equation> with $\mathbf E$ and <Faraday's law> with $\mathbf H$:
$$
\mathbf E\cdot\frac{\partial\mathbf D}{\partial t}
=\mathbf E\cdot(\nabla\times\mathbf H)-\mathbf E\cdot\mathbf J,
$$
$$
\mathbf H\cdot\frac{\partial\mathbf B}{\partial t}
=-\mathbf H\cdot(\nabla\times\mathbf E).
$$
Adding and using the vector identity
$$
\nabla\cdot(\mathbf E\times\mathbf H)
=\mathbf H\cdot(\nabla\times\mathbf E)
-\mathbf E\cdot(\nabla\times\mathbf H)
$$
gives
$$
\boxed{
\mathbf E\cdot\frac{\partial\mathbf D}{\partial t}
+\mathbf H\cdot\frac{\partial\mathbf B}{\partial t}
+\nabla\cdot(\mathbf E\times\mathbf H)
=-\mathbf E\cdot\mathbf J}.
$$
This is the local <Poynting theorem> before specializing the constitutive relations.
Back to article page
