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:
where ∇•S is the divergence of the Poynting vector (energy flow) and J•E 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:
in which B is the magnetic flux density. Using the divergence theorem, Poynting's theorem can be rewritten in integral form:
where is the boundary of a volume V. The shape of the volume is arbitrary but fixed for the calculation.
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:
- ε0 is the electric constant and μ0 is the magnetic constant.
- is the density of reactive power driving the build-up of electric field,
- is the density of reactive power driving the build-up of magnetic field, and
- is the density of electric power dissipated by the Lorentz force acting on charge carriers.