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In Purcell's Electricity and Magnetism, page 32, it is stated that the potential energy $~U$ of a system of charges is given by (using cgs units):

$$U=\frac{1}{8\pi}\int_{\mathrm{Entire \\field}} E^{2} \,d v$$

The author also said that we can think of this energy as being stored within the field itself, and that it has a density equal to $~E^{2}/8\pi~$ per unit volume. If it's the case, then why can't we use this same integral and simply integrate over the volume we are interested in finding the energy stored within?

Here's what Mr. Purcell said on this

"Our accounting comes out right if we think of it as stored in space with a density of $~E^{2}/8\pi~$ in $\textrm{ergs/cm}^{3}$ . There is no harm in this, but in fact we have no way of identifying, quite independently of anything else, the energy stored in a particular cubic centimeter of space. "

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The formula is taken as a definition of EM energy in the context where Poynting theorem is interpreted as an expression of local conservation of energy. One could define the EM energy differently than by that formula, but still in agreement with the same Poynting theorem. There is infinity of different but valid definitions of EM energy in this sense.

Purcell stresses that there is no way to determine which choice is correct. For simplicity, the usual Poynting formulae are used, but other alternative formulae could be used as well if we agreed to a different convention.

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