Energy and the Magnetic Field
Last 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 about it since magnetic fields can be slippery.
Let’s consider a static magnetic field generated by a collection of circuits , each carrying a current . Recall that Gauss’ law for magnetism tells us that ; since space is contractible, we know that its homology is trivial, and thus must be the curl of some other vector field , which we call the “magnetic potential” or “vector potential”. Now we can write down the flux of the magnetic field through each circuit:
This electromotive force must be counterbalanced by a battery maintaining the current or else the magnetic field wouldn’t be static.
We can determine how much power the battery must expend to maintain the current; a charge moving around the circuit goes down by in potential energy, which the battery must replace to send it around again. If such charges pass around in unit time, this is a work of per unit time; since — the current — we find that the power expenditure is , or.
Thus if we want to ramp the currents — and the field — up from a cold start in a time it takes a total work of
which is then the energy stored in the magnetic field.
This expression doesn’t depend on exactly how the field turns on, so let’s say the currents ramp up linearly:
and since the fluxes are proportional to the currents they must also ramp up linearly:
Plugging these in above, we find:
Now we can plug in our original expression for the flux:
This is great. But to be more general, let’s replace our currents with a current distribution:
Now we can use Ampère’s law to write
We can pull the same sort of trick last time to make the second integral go away; use the divergence theorem to convert to
and take the surface far enough away that the integral becomes negligible. We handwave that falls off roughly as the inverse fifth power of , while the area of only grows as the second power, and say that the term goes to zero.
So now we have a similar expression as last time for a magnetic energy density:
Again, we can check the units; the magnetic field has units of force per unit charge per unit velocity:
while the magnetic constant has units of henries per meter:
Putting together an inverse factor of the magnetic constant and two of the magnetic field and we get:
or, units of energy per unit volume, just like we expect for an energy density.