Beau- due to a computer glitch on our server your excellent question was lost. I won't try to reproduce all your correct thoughts about how the work available from a magnet is limited by the work put in. I will try to answer the part about how to calculate the energy stored in a magnet. Feel free to follow up with any parts of your question that I didn't remember. Again- sorry about the glitch, which also wiped out my own draft answer.
There are two parts to the stored energy. The simpler part is in the field outside the magnet, the rest is inside the magnet.
Outside the magnet there's an energy density that goes as the square of the magnetic field B
. In CGS units it's particularly simple. The energy density (in ergs/cm3
) is B2
/8π, where B
is in Gauss.You get the energy from integrating the density.
Inside the magnet, things are a little more complicated. You want the Helmhotz free energy, F, rather than the bare energy, U. F, which is U-ST where S is entropy and T is absolute temperature, is the quantity that reflects the actual work input and potential work output. For a weakly magnetized material, the F density is B2
/8π χ, where χ is the magnetic susceptibility. For an actual strong permanent magnet, that expression is pretty rough, since χ isn't well defined.
If you just want to get a rough idea of the stored energy, you can come pretty close just using the external field and the volume of the magnet. The point of that sort of calculation would just be to serve as a reminder that there's really not much stored energy in the magnet. For example, a large field (3000 G) over 1000 cm3
would have about 4x108
ergs or 40 J. That's about 1/100,000 of a kWh.
(published on 09/28/11)