# Static electric and magnetic fields and energy.

1. Jul 29, 2013

### chingel

It is well known that all sorts of systems tend towards minimal potential energy. I was wondering if this applies to static electric fields also, i.e. is an electric field such that it's energy integrated over all space is minimal? For example if we have a bounded source free region and if the electric field on the boundary is defined, does the electric field inside have minimal energy of all possible configurations? With possible configurations I mean all such which follow the Gauss law, but not necessarily the other Maxwell's laws.

The same question for magnetic fields.

2. Jul 29, 2013

### torquil

Yes. Maxwell's equations can be obtained from the principle of stationary action. In the time-independent case, this reduces a minimisation of the energy, for the given boundary conditions.

3. Jul 29, 2013

### Jano L.

This is true for the electrostatic field in a region where there is no free charge. The possible configurations are any electrostatic fields that have the same potential on the boundary surface. The electric intensity on the surface is not fixed, and the fields do not have to obey the Gauss law. If the field obeys the Gauss law and the boundary conditions, it is already the solution with minimal energy.

EDIT: This is true for vacuum. If we have linear dielectric and define energy by

$$W = \int \frac{1}{2}\mathbf E\cdot\mathbf D dV$$
the minimum theorem holds too, but the possible fields have to have the same electric displacement $\mathbf D$ on the boundary.

Last edited: Jul 29, 2013
4. Jul 30, 2013

### chingel

Thanks for the answers. Does anyone have a link to a proof? I would be interested in reading it.