How to find the mass-energy in a certain field

[Positron137]

How to find the "mass-energy" in a certain field

I saw somewhere that for a charged particle of radius R, the method of finding the "mass-energy" in such an electrostatic field (caused by the charged particle is)

M = ∫E^2 dV, where E is the electric field of the particle, and the bounds of the integral are from R to infinity. Can someone justify why this integral is correct? Thanks!

P.S. What exactly does "mass-energy" mean?

 

[Nugatory]

I saw somewhere that for a charged particle of radius R, the method of finding the "mass-energy" in such an electrostatic field (caused by the charged particle is)

M = ∫E^2 dV, where E is the electric field of the particle, and the bounds of the integral are from R to infinity. Can someone justify why this integral is correct? Thanks!

P.S. What exactly does "mass-energy" mean?

"Mass-energy" is a somewhat confusing way of thinking about what's going on here, but it's not wrong.

First, an electrical field does store energy. This is basic classical E&M; the easiest way to get at the relationship between electrical field and stored energy is to consider two infinite parallel charged plates, see how much work it takes to change the distance between them and how the field changes as a result.

Second, and somewhat independent of this exercise in classical E&M, we have the mass-energy equivalence implied by special relativity: $E=mc^2$, which allows us to talk about the combined mass-energy of a system - "total energy" might be clearer, as the mass is just another way of storing energy and vice versa.

 

[Positron137]

Ok. That makes much more sense. Thanks!