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Are you talking about the famous derivation of the displacement current, where Ampère's law is both true and false depending on what surface you choose to integrate through, despite the same boundary, as below: (Image from WikiMedia commons http://commons.wikimedia.org/wiki/File:Displacement_current_in_capacitor.svg) The solution of this was to add a ...

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If no charge is moving, there is no magnetic field. A point charge at rest has only an electric field, from "its" point of view. However, electric and magnetic fields are not seperate, since someone moving with respect to the resting charge would see a magnetic field due to the behaviour of the fields under Lorentz transformations. You may (for some ...

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Hints: This is related to the Maxwell action, cf. e.g. this and this Phys.SE posts. The EL equation is Ampere's law ${\bf \nabla}\times {\bf H}_0={\bf J}_0$, where ${\bf H}:=\frac{\bf B}{\mu}$ and ${\bf B}:={\bf \nabla}\times {\bf A}$. A necessary condition for the existence of a ${\bf H}_0$ solution to Ampere's law is the (stationary) continuity equation ...

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As most people know, "let there be light" is a famous biblical quote, from Genesis. Now, on to the teacher's shirt. Those equations on his back are Maxwell's equations. "Let there be light" is a joke, because Maxwell's equations describe electromagnetic fields, and light is a form of electromagnetic radiation, so the equations can be used to describe ...

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Disclaimer: This is answer is given from a mathematical physics point of view, and it is a little bit technical. Any comment or additional answer from other points of view is welcome. The classical limit of quantum theories and quantum field theories is not straightforward. It is now a very active research topic in mathematical physics and analysis. The ...

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I'll try to make a simple derivation. Suppose you have a unit point charge located at position $\vec{r}'$. Then the associated charge density is $\delta(\vec{r}-\vec{r'})$, which is a Dirac distribution. The electrostatic potential produced by this charge is given by the Coloumb's law: G(|\vec{r}-\vec{r}'|) = ...

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