Is charge energetic? The mass of a body is known to have two important features.


*

*It responds to and can be the source of a field (gravity).

*It is energetic ($E=mc^2$).
The charge of body is known to have at least one of these properties,
namely:


*

*It responds to and can be the source of a field (the EM-field).


I am curious if it could also have the other property as well. That is, could there be a charge equivalent of $E=mc^2$? Perhaps something like  
$$ E=Q\sqrt{\dfrac{c^4}{4 \pi G \epsilon_{0}}}~? $$
 A: I believe it should be
2) it is charged
Because charge itself is the analogue to energy for the EM field. Both energy and charge are conserved because of a deeper symmetry in the universe. For energy, this is the time symmetry of the laws of nature. For charge, this is the underlying symmetry of the EM field equations. See Noether's Theorem for more about this. In other words, charge itself is not energetic, it is another fundamental property.
$E=mc^2$ relates energy and mass. So to have something similar for EM, you'd be looking for an "electric mass", basically $q/c^2$. That's a very interesting idea. My best guess is that the EM field is too simple to allow something like that.
A: A field certainly indicates there is energy. However, the location of that energy is not in the field. For example, if I have two oppositely charged particles some distance apart, and then I allow them to move toward each other, I can extract mechanical energy from that process. So we know that two separated charges must have had energy. But the location of that energy is in the system rather than around each charge. If we look at only one of our two particles we can't tell where the energy is. Since energy in a closed system is called 'mass', The center of mass of our two particle system is exactly between the two particles assuming they are in a closed system. Of course, you can't fully isolate the two particles, though. (Note: It is energy that gravitates, and not anything else. The stress-energy tensor is simply a way of accounting for certain extra energy that would otherwise have been neglected).
