Energy stored in electric field

Question

I know that energy stored in electric field / unit volume = $\frac{1}{2} \epsilon\,E^2$.

so can I say that for any configuration calculating $\int \frac{1}{2} \epsilon\, E^2 \,d^3r$ over whole space - gives me the potential energy stored stored in the system

for instance for $Q_1$ and $Q_2$ ....separated by distance $d$,

$\int \frac{1}{2} \epsilon\, E^2 \,d^3r$ over whole space $= \frac{KQ_1Q_2}{d} $?

Possible duplicates: http://physics.stackexchange.com/q/78112/2451 and links therein. — Qmechanic, Apr 16 '16 at 15:16
The identity $\int \frac{1}{2} \epsilon, E^2 ,d^3r$ over whole space $= \frac{KQ_1Q_2}{d} $ is evidently untenable as the right-hand side may have negative sign whereas the left-hand side is always positive. — Valter Moretti, Apr 16 '16 at 15:23

score 1 · Answer 1 · answered Apr 16 '16 at 14:53

1

You will get infinity because in addition to $kQ_1Q_2/d$, it also includes the self-energy of the two point charges, which is infinity.

answered Apr 16 '16 at 14:53

velut luna

4,004

Care to expand. How infinity came here? – Anubhav Goel Apr 16 '16 at 14:57
Close to a charge, $E \sim 1/r^2$ and hence $E^2 \sim 1/r^4$. In $n$ dimensional space, $\int_{||x||<\epsilon} ||x||^{-\alpha}d^n x$ blows up when $\alpha>n$. – velut luna Apr 16 '16 at 15:02
I didn't say that. The OP said two point charges, not capacitor. In the case of point charges, the integral does diverge. – velut luna Apr 18 '16 at 04:41

Energy stored in electric field

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