Energy and the Electric Field
Okay, now let’s consider the electric field from the perspective of energy. We have an idea that this might be interesting because we know that the field produces a force, and forces and energies interact in interesting ways.
So recall that if we have a “test charge” at a point in an electric field it experiences a force . As we saw when discussing Faraday’s law, for a static electric field we can write for some “electric potential” function . Thus we can also write for the potential energy function .
Now, say the field is generated by a charge distribution ; how much potential energy is contained in the force the field exerts on the little bit of charge at ? We count , but this is too much — half of it is due to the rest of the distribution acting on the bit of charge at and half of it comes from acting back. We can thus find the total potential energy by integrating
Now, Gauss’ law tells us that , so we substitute:
Next we use a form of the product rule — — and run it backwards to write:
where we evaluate the first integral over space by evaluating it over the solid ball of radius and taking the limit as goes off to infinity. The divergence theorem says we can write:
where, as usual, we have taken the charge distribution to be compactly supported, so as our sphere gets large enough, the potential energy goes to zero. Yes, this is very hand-wavy, but this is how the physicists do it.
Anyway, what does this tell us? It means that a static electric field contains energy with a density
which we can integrate over any region of space to find the electrostatic potential energy contained in the field.
We can also check the units here; the electric field has units of force per unit charge:
while the electric constant has units of farads per meter:
Putting these together — two factors of and one of we find the units:
Joules per cubic meter — energy per unit of volume, just as we’d expect for an energy density.