Constant Electromagnetic Field Meaning 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.
 A: 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...).
A: 
How can constant electromagnetic field be explained in a "photon language"?

For a charged capacitor there is an electric field between the plates. In detail, there are electrons moved from one plate to the other plate and by this the intrinsic property of electric charge for charged subatomic particles is obvious as a macroscopic effect.
In permanent magnets there is an alignment of another intrinsic property of subatomic particles, their magnetic dipole moment (which is another word for their magnetic field). The same result of a macroscopic magnetic field one get if accelerate electrons. Moving electrons in circles show the same result as permanent magnets, their magnetic dipole moments get aligned and a macroscopic magnetic field occurs.
But for accelerated electrons also the phenomenon of electromagnetic radiation takes place. Means the accelerated electrons emit photons which have oscillating electric and magnetic field components. To stick it together, electrons have an electric field, a magnetic field and emit EM radiation. This is why the theory talks about virtual exchange of photons for interactions of electric fields as well as of magnetic fields. But in your case of a static field it seems not to be necessary to use the image of virtual photons. 
