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I was reading Does Electric field has mass Energy; it gives energy density k stored around an electric charge as $ k = \dfrac{q^2}{r^4}$ in skeleton form without any constants.

So, a definite amount of energy is stored around an electric charge. We can extract it by moving electric charges in it.

So, it should mean there should be a limit to amount of energy we can extract, since energy stored is limited?

Am I correct?

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The equation you provided is actually given by

$$W=\frac{\epsilon_0}{2}\int E^2\,\mathrm d\tau$$

which is the energy stored in an electric field. This energy is utilized by the charge to generate it's field of influence or it's electric field. It's dependent on the magnitude of charge and the distance of separation between the charge and the point of concern.

When you place another charge in it's vicinity (say to extract the energy as you said), the charge get accelerated by the field and an accelerated charge will radiate energy as electromagnetic waves. These electromagnetic waves carry energy (and that includes magnetic energy term also). Of these, due to the acceleration of the charge, some of it's energy is radiated and the rest is stored in the fields. This is Poynting's theorem. The radiated energy will not come back. It will radiate out to infinity. So you will be left only with what's stored in the field. That will continue to decrease as the wave propagates. So there is always a limit up to which you can extract energy from the field. Remember that the field itself is a driving energy. So you could do some work with it, thereby extracting it's energy as work. But that also will be limited. The equation will not be applicable near the charge as the electric field contains a 1/$r^2$ term that blows up near the charge. So, does that mean the energy stored by a point charge when integrated over the all space gives you infinite energy? We know that it is practically not possible. So in this case, we cannot treat our charge as a point entity even though we say in the case of an electron (because we don't know any further substructure of an electron). It's actually a point entity, a point source of electromagnetic energy, with energy density infinite at the point like a huge black hole singularity. That may be the better way to see a charge.

The energy can be extracted from an electric field, which is made used in a battery. But anything that could extract that energy should have a characteristic property called charge. Once you place a charge in it's vicinity, it will accelerate, which means you cannot use the entire energy from it, only a part of it.

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  • $\begingroup$ I understand why energy cannot be ∞ near charge, since in physical world no point charge exist, it always occupies some volume. So, at surface energy stored will be max and further decrease towards center. $\endgroup$ Commented Apr 23, 2016 at 3:05
  • $\begingroup$ $$W=\frac{\epsilon_0}{2}\int E^2\,\mathrm d\tau$$ When we integrate it for an electron, we can take radius of electron as lower limit and ∞ as upper. Will this give me max energy I can extract? $\endgroup$ Commented Apr 23, 2016 at 3:12
  • $\begingroup$ No, if you extract that much energy, then there will be no electron there. The electron itself exists as that energy. Also the electron has no radius. That's cannot be. Electron is a structureless point entity. It's a fundamental particle $\endgroup$
    – UKH
    Commented Apr 23, 2016 at 8:48
  • $\begingroup$ alternativephysics.org/book/ElectronStructure.htm $\endgroup$ Commented Apr 23, 2016 at 9:01
  • $\begingroup$ In classical physics we can roughly estimate it. If you still insist, let's take a proton which has known radius. $\endgroup$ Commented Apr 23, 2016 at 9:03

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