The electric field between two capacitor plates is very simple.
$$
\vec{E} = \frac{Q}{\epsilon_0 A} \vec{e}_z
$$
I can get the energy stored in the field by integrating the energy density, $u_e$, over the volume (between the plates).
$$
U = \int_V u_e \; \text{d}^3\!x = \int_V \frac{\epsilon_0}{2} E^2 \; \text{d}^3\!x
$$
Since the field is constant, if I pull the plates appart—say that I double the distance—the integration volume is now twice what is was, and the energy stored in the field doubles. Fine!
Simultaneously, we can make an argument from potential energy of the charges in the plates. The charges in each plate are attracted to the other, so when I pull them appart there is a force, and I'm doing work which gives the charges additional potential energy; in virtue of their increased separation.
My question is: Are these two separate processes, where energy stored in the field AND in the potential energy of the charges. Or, are these two different ways of describing the same physical fact that the energy of the system is increasing?
Cheers!
Best Answer
As Griffiths has said. It is simply a matter of book keeping on whether or not you would like to say that the collection of charges has an associated potential energy to it. Or you would like to say that the E field possesses some energy density. It is the exact same thing, and yes 2 different ways of describing it! Look up the derivation of field energy ( in griffiths). You start with the formula for the potential energy of a general charge distribution, then use maxwells equations to eliminate ρ in favour of the fields!