Poynting's theorem

In electrodynamics, Poynting's theorem is a statement of conservation of energy for the electromagnetic field, in the form of a partial differential equation, due to the British physicist John Henry Poynting. Poynting's theorem is analogous to the work-energy theorem in classical mechanics, and mathematically similar to the continuity equation, because it relates the energy stored in the electromagnetic field to the work done on a charge distribution (i.e. an electrically charged object), through energy flux.

In words, the theorem is an energy balance:

A second statement can also explain the theorem - "The decrease in the electromagnetic energy per unit time in a certain volume is equal to the sum of work done by the field forces and the net outward flux per unit time".

Mathematically, this is summarised in differential form as:

${\displaystyle -{\frac {\partial u}{\partial t}}=\nabla \cdot \mathbf {S} +\mathbf {J} _{f}\cdot \mathbf {E} }$

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