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.
[…] time we calculated the energy of the electric field. Now let’s repeat with the magnetic field, and let’s try to be a little more careful […]
Pingback by Energy and the Magnetic Field « The Unapologetic Mathematician | February 14, 2012 |
[…] in sight! On the left, we’re taking the derivative of the combined energy densities of the electric and magnetic […]
Pingback by Conservation of Electromagnetic Energy « The Unapologetic Mathematician | February 17, 2012 |
dV has been subscripted in the formula about halfway down.
About the hand-wavy part: but the surface area goes to infinity, so is there reason to believe that it’s true?
Thanks for catching that; I fixed a few places it had gotten dropped entirely as well.
As for the hand-wavy bit, basically if you get far enough from a compact charge distribution it looks like a point charge, so
falls off as the square of the distance,
falls off as the fourth power, and the surface area only goes up as the square of the distance.