# Poynting's theorem

In electrodynamics, Poynting's theorem is a statement of conservation of energy for the electromagnetic field,[clarification needed] in the form of a partial differential equation, due to the British physicist John Henry Poynting.[1] 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.

## Statement

### General

In words, the theorem is an energy balance:

The rate of energy transfer (per unit volume) from a region of space equals the rate of work done on a charge distribution plus the energy flux leaving that region.

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:

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

where ∇•S is the divergence of the Poynting vector (energy flow) and JE is the rate at which the fields do work on a charged object (J is the current density corresponding to the motion of charge, E is the electric field, and • is the dot product). The energy density u is given by:[2]

${\displaystyle u={\frac {1}{2}}\left(\epsilon _{0}\mathbf {E} \cdot \mathbf {E} +{\frac {1}{\mu _{0}}}\mathbf {B} \cdot \mathbf {B} \right)}$

in which B is the magnetic flux density. Using the divergence theorem, Poynting's theorem can be rewritten in integral form:

 ${\displaystyle -{\frac {\partial }{\partial t}}\int _{V}udV=}$ ${\displaystyle \scriptstyle \partial V}$ ${\displaystyle \mathbf {S} \cdot d\mathbf {A} +\int _{V}\mathbf {J} \cdot \mathbf {E} dV}$

where ${\displaystyle \partial V\!}$ is the boundary of a volume V. The shape of the volume is arbitrary but fixed for the calculation.

### Electrical engineering

In electrical engineering context the theorem is usually written with the energy density term u expanded in the following way, which resembles the continuity equation:

${\displaystyle \nabla \cdot \mathbf {S} +\epsilon _{0}\mathbf {E} \cdot {\frac {\partial \mathbf {E} }{\partial t}}+{\frac {\mathbf {B} }{\mu _{0}}}\cdot {\frac {\partial \mathbf {B} }{\partial t}}+\mathbf {J} \cdot \mathbf {E} =0,}$

where

• ε0 is the electric constant and μ0 is the magnetic constant.
• ${\displaystyle \epsilon _{0}\mathbf {E} \cdot {\frac {\partial \mathbf {E} }{\partial t}}}$ is the density of reactive power driving the build-up of electric field,
• ${\displaystyle {\frac {\mathbf {B} }{\mu _{0}}}\cdot {\frac {\partial \mathbf {B} }{\partial t}}}$ is the density of reactive power driving the build-up of magnetic field, and
• ${\displaystyle \mathbf {J} \cdot \mathbf {E} }$ is the density of electric power dissipated by the Lorentz force acting on charge carriers.