For electrostatics one can work out the energy of a charge configuration in terms of its associated field:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

U = \epsilon_0 \vec{E}^2/2

[/tex]

I've seen that the general energy of the field is given by:

[tex]

U = \epsilon_0 (\vec{E}^2 + c^2\vec{B}^2)/2

[/tex]

but am unsure how to show this. I suspect that a relativistic argument would be one way.

Another way, which doesn't work (or I did it wrong), is a calculation of the Hamiltonian for the (non-proper time formulation) of the Lorentz force Lagrangian. I get an energy that only includes the q\phi part as in electrostatics. This actually makes some sense since doing a line integral against a perpendicular field won't contribute.

I'd be interested to know two things:

1) A high level non-math description of where the B^2 term comes from.

2) Some hints for the math side of the question. What is a way (or some ways) that the E^2 + B^2 form of the field energy can be derived?

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# Electric and magnetic field energy

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