Poynting theorem
= Poynting theorem
{c}
{title2=$\partial_tu+\nabla\cdot\mathbf S=-\mathbf J\cdot\mathbf E$}
= Poynting's theorem
{c}
{synonym}
The <Poynting theorem> is the local <energy> balance of classical <electromagnetism>. For vacuum field laws, with <current density> $\mathbf J$, the <electromagnetic energy> per unit volume is
$$
u=\frac{\epsilon_0|\mathbf E|^2}{2}+\frac{|\mathbf B|^2}{2\mu_0},
$$
and
$$
\partial_tu+\nabla\cdot\mathbf S=-\mathbf J\cdot\mathbf E.
$$
Thus a decrease in local <electromagnetic energy> can supply outward <energy flux> or <work> on <electric charge>. The term $\mathbf J\cdot\mathbf E$ is <power> transferred to matter per unit volume. Its sign matters: positive values remove field <energy>, while negative values describe matter supplying the field. The magnetic part of the <Lorentz force> supplies no work term.
For a fixed volume $V$, the <divergence theorem> gives the directly measurable balance
$$
\frac{d}{dt}\int_Vu\,dV
=-\int_{\partial V}\mathbf S\cdot\mathbf n\,dA
-\int_V\mathbf J\cdot\mathbf E\,dV.
$$
For example, $2.0\,\mathrm W$ entering a region with $0.50\,\mathrm W$ leaving and $1.20\,\mathrm W$ transferred to matter means stored <electromagnetic energy> increases at $0.30\,\mathrm W$. In source-free vacuum only the boundary flux changes the stored <energy>; a nonzero local <Poynting vector> can still have zero net closed-surface flux.
A brief proof takes the <dot product> of the <Ampère-Maxwell equation> with $\mathbf E/\mu_0$ and uses <Faraday's law> with $\mathbf B/\mu_0$. The identity $\nabla\cdot(\mathbf E\times\mathbf B)=\mathbf B\cdot(\nabla\times\mathbf E)-\mathbf E\cdot(\nabla\times\mathbf B)$ turns the remaining terms into the <divergence> of the <Poynting vector>. https://www.feynmanlectures.caltech.edu/II_27.html#Ch27-S3[Feynman lectures, II.27.3: local field-energy conservation]{ref}.