### 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

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!

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!

Labels: electricity, electromagnetism, magnetism, physics, science

## 12 Comments:

Cool post. Nice to see science presented in a friendly manner.

I remember when I was first introduced to work as F·s and being told that the result was a vector. I was rather disheartened to learn that if an object (e.g. me) goes somewhere (e.g. work) from somewhere (e.g. home), and them back again, no work has been done. I honestly thought I wouldn't get paid at whatever job I had when I was older.

Well, I was young!

And I really need to get my hands on some neodymium magnets. I keep hearing people say how cool they are, and I want to play! I bet someone in one of the labs has some...

By nullifidian, at 8:40 AM, April 25, 2007

nillifidian, you can get them online pretty easily, try United Nuclear (I think that's the name, not enough time to look it up).

And that only works in a conservative field without friction, so you do plenty of work getting to and from work.

By Stupac2, at 8:42 AM, April 25, 2007

Junk hard drives also tend to contain NdFeB magnets. Be careful of sharp bits and other hazards, especially pinching your fingers between the magnets. They're shaped funny, and contained in iron brackets to direct the field...shouldn't be too hard to find, if you want to play.

By Joel, at 2:57 PM, May 04, 2007

Very nice description.

By Kurt, at 9:16 AM, July 30, 2007

"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."

Not true. f(x)=1/x falls to zero as the distance away increases, but the integral from 0 to infinity of f(x)dx IS infinite. And there are some configurations of magnetic objects that follow a 1/r curve, like the field produced by a long, straight wire.

Also, if magnetic fields do no work, then where does the energy come from? If I place two magnets side by side as such:

|N-S| |N-S|

they will accelerate toward each other, which indicates a force pushing them directly together. Since the force and the distance moved are in the SAME direction, some work is being done. Where does that energy come from?

By Phil, at 5:54 PM, February 27, 2008

Phil, magnetic fields (like most fields) fall of like 1/r^2, hence the so-called inverse-square law. So the integral over the field would be finite. Besides, energy is actually the square of the field, so it would fall off even faster.

Like I said in the post, the work done comes from dissipating the field. There's no way to harvest energy from a magnetic field except by making the field weaker, and in order to make the field stronger energy must be put back in. That's what you're doing when you let magnets come together then pull them apart, changing the field. But the field itself isn't doing any work, just changing based on interactions with the surroundings.

By Stupac2, at 6:23 PM, February 27, 2008

"Not true. f(x)=1/x falls to zero as the distance away increases, but the integral from 0 to infinity of f(x)dx IS infinite. And there are some configurations of magnetic objects that follow a 1/r curve, like the field produced by a long, straight wire."

This field uses the approximation of a wire of infinite length which, whilst very good, is not strictly true. The philosophical problem is then moving an infinite distance from something of infinte length, as inifinte distance usually means to the point where an object is point-like, which if it is infinitely long it cannot become.

So basically this isn't quite true because the formula for the field isn't quite correct.

By Anonymous, at 4:11 PM, April 18, 2008

Proof is ill defined in the 'real' world. Also just saw that just after the "W'right" brothers flew, Scientific American proved mathematically that human flight was impossible.

"Free" energy depends on the relative position of your opinion. Like po-tat-o and po-tay-to, I never paid a cent for solar energy that warms my bones, but then again I always get back to being cold again, but it still "works".

You can create a system that puts a counter on product to charge and see that entropy puts it back out of order. Doesn't mean it didn't work at some time.

Look quick, don't be caught dead seein'.

By hal2009, at 1:13 AM, September 01, 2009

about what Phil said

"Also, if magnetic fields do no work, then where does the energy come from? If I place two magnets side by side as such:

|N-S| |N-S|

they will accelerate toward each other, which indicates a force pushing them directly together. Since the force and the distance moved are in the SAME direction, some work is being done. Where does that energy come from?"

the energy comes from you pulling the magnets apart. you can comepare it to gravity: it you hold an object above the ground and let it go, it will fall by itself. the energy comes from you first lifting the object, thus giving it potential energy.

By John, at 1:55 PM, August 28, 2010

Damn! I was looking for Magnetic Fields on Google, but I didn'0t think it was so complicated! Dude, are you a scientist or something like that?

cheap viagra Still, thank you for the info, at least I think I got the main idea of it!

By Mark, at 11:09 AM, April 28, 2011

This can't truly have success, I suppose so.

By www.granada-3d.com, at 2:48 AM, October 14, 2011

Well, I do not actually imagine it may work.

By www.albacete-3d.com, at 9:24 AM, November 23, 2011

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