How can constant electromagnetic field be explained in a "photon language"?
Suppose we have a capacitor in which $\textbf{E}$ holds constant. I am having hard times applying the idea of photons to this situation.
Answer
You can't explain a constant electromagnetic field (or, for that matter, any electromagnetic field that takes a definite value) in "photon language".
The concept of photons is inextricably linked with quantum field theory. In quantum field theory, the electromagnetic field is not an object that takes a definite value, it is an operator acting on quantum states. You can examine its expectation value at particular points, but actually formulating all actually occuring electromagnetic fields in this fully quantum language is infeasible, in particular since it's pretty difficult to put definitely localized particles anywhere in relativistic quantum field theory - and there is no good non-relativistic version of the photon.
Things like a constant electric field, or most electromagnetic fields you will ever encounter in usual applications, gain nothing from knowing the underlying quantum field theory with its photons. In the classical non-relativistic limit of quantum electrodynamics, one can derive Coulomb's law between charges, and re-introducing special relativity then gives us Maxwell's equations, aka classical electromagnetism. See this and this question for the respective derivations. It is in this fully classical framework, which knows nothing of photons, that most electromagnetic fields are usefully described. The full quantum description is much too complicated and rarely allows additional or more precise predictions in settings where the classical limit is appropriate (which is kind of tautological because it's appropriate precisely when the quantum description doesn't add much...).
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