I assume I am making a mistake here. Can you please help me learn how to fix them?(adsbygoogle = window.adsbygoogle || []).push({});

In electrodynamics, the gauge transformations are:

[tex]\vec{A} \rightarrow \vec{A} + \vec{\nabla}\lambda[/tex]

[tex]V \rightarrow V - \frac{\partial}{\partial t}\lambda[/tex]

These leave the electric and magnetic fields unchanged. The electromagnetic energy density is proportional to:

[tex] u \propto (E^2 + B^2)[/tex]

So the energy density should be gauge invariant. A guage transformation shouldn't even shift it by a constant.

However, the energy density can also be written in terms of the potentials and charge distributions as:

[tex] u = \frac{1}{2}(\rho V + \vec{j} \cdot \vec{A})[/tex]

If I let [itex]\lambda[/itex] = e^(-r^2), then V is unchanged, and A is just changed by a radial field which vanishes at infinity. Here, the energy density IS changed, and not by a constant amount either ... it changes by an amountdependingon j. What gives?

Even weirder, is some textbooks start with the potentials form toderivethe fields form. I don't see any place they fix any gauge in doing such derivation. Please help.

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# Electromagnetic energy is not Gauge invariant?

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