Magnetic Fields Do No Work

Every time I head about a perpetual motion machine (or some other kind of free energy), I know immediately how it works. It’s one of a short list of well-known but poorly-understood (at least among the general population) physical principles. The two big ones are magnetic fields and the Casimir Effect. I’m going to be talking about magnetic fields today, and why they can never be used to make any kind of perpetual motion machine.

Everyone loves magnets, and if they don’t they should. I know I do, I have about 40 neodymium super-magnets on my desk. They’re fascinating, and with their invisible and seemingly magical attraction, they have mystified child and adult alike for generations. Surely there must be some way to harness this power and get free energy!

There’s just one problem with this: magnetic fields do no work.

Some physics background for those who don’t have it, work is basically force times distance. If you apply a force over a given distance you do work, any time you move an object you’re doing work (however, sitting at your computer and playing solitaire is not work). Work has the units of energy, and (ignoring friction) the work done moving an object is exactly equal to the change in its energy.

So if magnetic fields don’t do work, then we can’t get any energy out of them without dissipating the field itself.

But, you ask, how do I know that magnetic fields don’t do work? The answer to that requires some vector calculus (unfortunately, no one likes vector calculus), but it’s not too bad. Skip it if you don’t care, but I promise I’m not going to kill you with Math.

The magnetic field (B) is defined as:



That X in the middle does not mean “times”, it’s the cross product, which basically means that the force from a moving charge in a magnetic field is perpendicular to both the field and to the velocity of the charge.

Work, in the true mathematic form, is:



F is the force and ds is a bit of the path, “dotted” into the force. But we know that ds is equal to the velocity time a small bit of time, dt (because that’s the part of the path that the object moves in time dt).

Now recall that the force is equal to the field crossed with the velocity, and to get work we have to dot it with the velocity:



The cross and the dot products have a peculiar property that if this happens, the result is always zero. Geometrically this happens because the cross product creates a vector that is perpendicular to both the initial vectors, but the dot product evaluates the length that two vectors have in common. If they’re perpendicular, then the answer is always zero.

So magnetic fields do no work, and hence you can’t get any energy from them (without dissipating the field).

If you don’t like that argument (although it’s perfectly solid), I’ve got another one for you. The energy density of the magnetic field is:



If you integrate that over all space, you get the entire amount of energy contained in the field. Because every magnetic field falls to zero as the distance away increases, that integral is finite, and hence the total amount of energy one can extract from a field is finite. This is why I repeatedly added “without dissipating the field” to the end of “magnetic fields do no work.” This is also why objects (such as paper clips) will go flying towards magnets: they modify the field, changing how much energy is stored in it. You could conceivably extract energy from this, but only as much as you put into it (and actually less because of losses due to friction), just like every other physical system.

So, while magnets are fun and fascinating, anyone who claims that they’ve harnessed free energy from them is either mistaken or a liar, and they have an incomplete grasp of electromagnetism. Remember, magnetic fields do no work!