# Stored Energy in a Magnet

*Most recent answer: 09/27/2011*

Q:

How do you calculate the stored energy in a permanent magnet?

- Beau (age 20)

- Beau (age 20)

A:

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/cm

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 B

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 cm

Mike W.

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/cm

^{3}) is B^{2}/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 B

^{2}/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 cm

^{3}would have about 4x10^{8}ergs or 40 J. That's about 1/100,000 of a kWh.Mike W.

*(published on 09/27/2011)*

## Follow-Up #1: magnet stored energy

Q:

I read your answer about calculating the stored energy in a magnet, at the end you summarised that a cubic metre magnet, 3000 G would have about 1/100,000 kw. Approx. Could the math or definitions be wrong? My understanding is that energy is the ability to do work. If thats so, I have a magnet thats only 50mm x 50mm x 50mm. Its currently holding up a weight of 50kg and has been doing so for over a year. I wonder how much power would be required to do the same with an electro magnet? Just for a minute let alone a year or more. No battery I know of could sustain that load for that long, and even a 12 volt car battery couldn't do that and that I am sure has more than 1/100,000 kw. Im not university trained but something doesn't seem right with the math, or maybe the math is right but the application of it was wrong? Please help me to understand.

- Pablo Saavedra (age 44)

Preston,Victoria,Australia

- Pablo Saavedra (age 44)

Preston,Victoria,Australia

A:

That''s about 100,000 kW-hr! A kW is a unit of power (rate of energy change) not an energy.

This issue is that it doesn't take any work to hold a weight at fixed height. If you set the weight on a table, does the table need some energy source to hold it up? Does a car need to burn fuel to keep from sliding down if it's parked on a slight slope? The energy just stays fixed, with no flow.

Here's an old thread where we discuss these issues at some greater depth: http://van.physics.illinois.edu/qa/listing.php?id=339.

Mike W.

*(published on 06/30/2016)*