Static electric and magnetic fields and energy.

AI Thread Summary
Static electric fields tend to configurations that minimize potential energy, particularly in bounded, source-free regions where boundary conditions are defined. The electric field inside such regions can achieve minimal energy if it adheres to Gauss's law, although it does not need to comply with all of Maxwell's equations. Similarly, magnetic fields also follow this principle of energy minimization under stationary action conditions. In vacuum, electrostatic fields with the same boundary potential represent the minimal energy solution. For linear dielectrics, the minimum energy theorem applies as long as the electric displacement on the boundary remains consistent.
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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.
 
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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.
 
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.
 
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Thanks for the answers. Does anyone have a link to a proof? I would be interested in reading it.
 
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