# What's more important, a magnet's surface area or thickness?

November 5, 2009 11:27 AM Subscribe

What contributes more to the attractive force between a magnet and a thin piece of metal - the magnet's surface area, or its thickness?

Suppose I have thin piece of metal and a neodymium disc magnet. With everything else being equal, will a thin magnet with a large surface area provide a stronger pulling force than a thicker magnet with a smaller diameter? In either case, the piece of metal is no thicker than the magnet itself.

Suppose I have thin piece of metal and a neodymium disc magnet. With everything else being equal, will a thin magnet with a large surface area provide a stronger pulling force than a thicker magnet with a smaller diameter? In either case, the piece of metal is no thicker than the magnet itself.

I should be able to do this problem, but my E&M is rusty. So I'll handwave.

Suppose that the magnet's field induces some dipole moment in the piece of metal (this is why you get attraction between a magnet and a fridge). Then we can imagine dipoles distributed throughout both volumes. The energy of two dipoles is a strong function of 1/r, and since many of those dipoles are much farther away in the thick magnet, the total energy is smaller in the thick / small diameter pair. The force is like the derivative of the energy, and I assume (this is the major weak point in the argument) that they both end up scaling with the same powers, just different leading factors.

This suggests that: in the limit that thickness >> width, the force is weaker.

posted by gensubuser at 12:33 PM on November 5, 2009

Suppose that the magnet's field induces some dipole moment in the piece of metal (this is why you get attraction between a magnet and a fridge). Then we can imagine dipoles distributed throughout both volumes. The energy of two dipoles is a strong function of 1/r, and since many of those dipoles are much farther away in the thick magnet, the total energy is smaller in the thick / small diameter pair. The force is like the derivative of the energy, and I assume (this is the major weak point in the argument) that they both end up scaling with the same powers, just different leading factors.

This suggests that: in the limit that thickness >> width, the force is weaker.

posted by gensubuser at 12:33 PM on November 5, 2009

The pulling force is (IIRC, and I'm pretty vague on this subject) proportional to the square of the magnetic flux density times the area (B

If this is a free magnet near a steel sheet (eg a fridge magnet with no pole piece) then the air gap

posted by hattifattener at 12:45 PM on November 5, 2009

^{2}A), assuming that the air gap dominates the circuit's magnetic resistance. A thick magnet will have a proportionally higher mmf which means a proportionally higher B. A magnet of the same thickness (which I'm simplistically equating to mmf per area) but larger area will have the same B but larger A. So, if you're allowed to double either the thickness or the area of the magnet, you'll get more benefit from a thicker magnet.If this is a free magnet near a steel sheet (eg a fridge magnet with no pole piece) then the air gap

*doesn't*dominate the circuit; the path from the back of the magnet around to the nearby steel does. The math gets complicated, but I suspect that since the B-field isn't as dependent on the gap distance in this case, the attractive force goes more like B times A, in which case doubling the thickness vs. doubling the area give you about the same benefit. Also, sticking a modern supermagnet to a piece of steel probably saturates the steel near where the magnet is, which confuses the issue in another way. And I'm not even considering the fact that any real magnet (or piece of steel) has a nonlinear B-H curve.posted by hattifattener at 12:45 PM on November 5, 2009

Assuming the size of the magnet (really the number of atoms in it) is the same either way, you will get more force by positioning the magnet closer to the material.

Consider two cases, where T is the target and X is a magnetic material:

However, in a practical application there are some other things to consider: a "thin" magnet may spread the force out across a greater area on the target, which might be undesirable. Also, it's difficult to work with very thin rare-earth magnets; you might get to a point where, if you got it stuck to a ferrous material, it was impossible to remove without breakage. (I've never seen one this thin so there also might be manufacturing concerns.)

Also, if the "magnet" is really a coil of wire or something else besides a bunch of tiny magnetic dipoles, then its geometry may matter a lot, but that's a much more complex problem.

posted by Kadin2048 at 2:05 PM on November 5, 2009

Consider two cases, where T is the target and X is a magnetic material:

T T X XXX X XIf you look at the average distance from some infinitesimal piece of magnet to the target, the "thick" magnet is further away than the "thin" one.

However, in a practical application there are some other things to consider: a "thin" magnet may spread the force out across a greater area on the target, which might be undesirable. Also, it's difficult to work with very thin rare-earth magnets; you might get to a point where, if you got it stuck to a ferrous material, it was impossible to remove without breakage. (I've never seen one this thin so there also might be manufacturing concerns.)

Also, if the "magnet" is really a coil of wire or something else besides a bunch of tiny magnetic dipoles, then its geometry may matter a lot, but that's a much more complex problem.

posted by Kadin2048 at 2:05 PM on November 5, 2009

Response by poster: Thanks for the answers. They more or less verify what I expected.

echo target: I have a bunch of thin (1mm) neodymium magnets I want to use to mount some metal tins. One isn't enough to hold the weight, but 3 together appears to be. I'm trying to work out if I should get more of the small magnets and use them together, get larger diameter but still thin magnets, or get thicker magnets.

posted by howling fantods at 5:24 PM on November 5, 2009

echo target: I have a bunch of thin (1mm) neodymium magnets I want to use to mount some metal tins. One isn't enough to hold the weight, but 3 together appears to be. I'm trying to work out if I should get more of the small magnets and use them together, get larger diameter but still thin magnets, or get thicker magnets.

posted by howling fantods at 5:24 PM on November 5, 2009

This thread is closed to new comments.

Is this for a particular application?

posted by echo target at 11:53 AM on November 5, 2009