Poynting's Theorem

${\text{d}}_{t}W=-{\text{d}}_t{\int }_{\mathcal{V}}u\mathrm{d}\tau -{\oint }_{\mathcal{S}}\vec{S}\cdot \mathrm{d}\vec{a}$

• Work done on charges by fields is equal to decrease in energy of the fields plus the energy flowing out of the surface
• $u$ is the electromagnetic field energy density
• $\vec{S}$ is the Poynting vector