Maximized Magnets

Recently another post from Hack-a-Day caught my eye, discussing a technique for stacking magnets to get a stronger field, called a Halbach Array. You might assume that if you have a bunch of small magnets, the best way to combine them would be stack them with all their poles pointing in the same direction, but it turns out you can get a stronger field by stacking them in an unintuitive pattern:

Wikipedia

The arrows point from south to north pole of each magnet. Using the (wonderfully named) Python package Magpylib we can look at the magnetic field produced by this combination:

The field strength is represented by a higher density of lines, but that's a little hard to read off this plot, so we can also plot the overall magnitude of the field on a line just above and just below the magnets:

Not only is the field stronger than a single magnet's, it's stronger on one side than the other! This gave me pause, since it sounded like a monopole magnet, which is forbidden by Maxwell's Equations, specifically Gauss's Law for Magnets:

This says that for any closed surface S, all the field lines going out of the surface need to be matched with lines coming in, so that the sum cancels. It seems like our setup could have more lines going out the bottom than coming in the top, but the key is that even though the magnitude of the field is stronger on one side of the array, it includes both north and south poles. You can see this in the field plot above: there are lines going in and out on both sides. We can double check by integrating around a box as S:

When the loop closes, we're back to zero, and no laws are violated!

That last section may have been a little technical, so I'd like to end on something (I find) beautiful. In the Wikipedia article for Halbach arrays, they mention a version using magnetic rods, which can be rotated to switch the field from one side to the other. I was curious how the field strength varied during this, so I made an animation with the output from Magpylib:

The dots mark the north pole of each rod. Each 90° rotation swaps the strong side, but I think the movement of the 3 low-field nodes is really cool. Thanks, Hack-a-Day, for introducing me to this nifty structure!