Poynting's theorem

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Poynting's theorem is a statement due to John Henry Poynting about the conservation of energy for the electromagnetic field. It relates the time derivative of the energy density, u, to the energy flow and the rate at which the fields do work. It is summarised by the following formula

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

where S is the Poynting vector representing the flow of energy, J is the current density and E is the electric field. Since the magnetic field does no work, the right hand side gives the negative of the total work done by the electromagnetic field per second·meter³.

Poynting's theorem in integral form:

$\frac{\partial}{\partial t} \int_V u \ dV + \oint_{\partial V}\mathbf{S} \ d\mathbf{A} = -\int_V\mathbf{J}\cdot\mathbf{E} \ dV$

Where $\partial V \!$ is the surface which bounds (encloses) volume $V \!$ .

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:

$\nabla\cdot\mathbf{S} + \mathbf{E}\cdot\frac{\partial \mathbf{D}}{\partial t} + \mathbf{H}\cdot\frac{\partial\mathbf{B}}{\partial t} + \mathbf{J}\cdot\mathbf{E} = 0$

Where $\mathbf{S}$ is the energy flow of the electromagnetic wave, $\mathbf{E}\cdot\frac{\partial \mathbf{D}}{\partial t}$ is the power consumed for the build-up of electric field, $\mathbf{H}\cdot\frac{\partial\mathbf{B}}{\partial t}$ is the power consumed for the build-up of magnetic field and $\mathbf{J}\cdot\mathbf{E}$ is the power consumed by the Lorentz force acting on charge carriers.