Energy Density in Electric Field

Question

While studying the Capacitors, I read and understood that the Energy stored in a Capacitor which is given by (1/2)Q V.

The next topic was the Energy density and the book stated that the energy stored in a Capacitor is electrostatic potential energy.

When we pull the plates of a capacitor apart, we have to do work against the electrostatic attraction between the plates.

In which region of space is energy stored?

When we increase the separation between plates from $r_1$ to $r_2$ an amount $$Q^2/2 \epsilon_0 A (r_2-r_1)$$ of work is performed by us and this much energy goes into the capictor.In the other hand,new electric field is created in volume $A(r_2-r_1)$. We conclude that the energy is stored in this volume. This last part made no sense to me,the energy supplied is to do the work and the energy goes into the capacitor.

Much like the gravitational pull of the the earth attracts a particle if we increase its height by $h$ (potential increases) and the energy is stored in the body.

Similarly in this case,the energy should be stored inside the capacitor.

Can someone explain what is happening here and why is this not the same as in the case of Earth's gravitation?

Even when there are two small equal and opposite charges, when one charge is pushed away from the other, the potential of the charge increases and it is stored in the charge not in space. Someone explain?

Answer 1

Quoting Griffiths' Introduction to Electrodynamics:

Where is the energy stored? Equations 2.43 and 2.45 offer two different ways of calculating the same thing. The first is an integral over the charge distribution; the second is an integral over the field. These can involve completely different regions. For instance, in the case of the spherical shell (Ex. 2.9) the charge is confined to the surface, whereas the electric field is everywhere outside this surface. Where is the energy, then? Is it stored in the field, as Eq. 2.45 seems to suggest, or is it stored in the charge, as Eq. 2.43 implies? At the present stage this is simply an unanswerable question: I can tell you what the total energy is, and I can provide you with several different ways to compute it, but it is impertinent to worry about where the energy is located. In the context of radiation theory (Chapter 11) it is useful (and in general relativity it is essential) to regard the energy as stored in the field, with a density $$\frac{\epsilon_0}{2}E^2 = \text{energy per unit volume.}$$ But in electrostatics one could just as well say it is stored in the charge, with a density $\frac{1}{2}\rho V$. The difference is purely a matter of bookkeeping.

In radiation theory, it is indeed useful to think of the energy being stored in the field: Poynting's theorem then becomes a statement of conservation of electromagnetic energy. However, I don't know if there are more solid grounds for accepting this (a proof or experimental evidence) in general relativity: I hope someone else can comment on this.