I have attached a plot of ‘magnetic potential’ for our ‘monopole cube’ just for completeness. Boring plot, isn’t it. Outside of the cube is black, meaning the magnetic potential is uniformly zero. Inside of the cube, the magnetic potential is very high at 1782 ampere-turns (yellow colour), but equally uniform. The uniformity of magnetic potential tells us that there will be no flux ‘flowing’ from one place to another, and thus the flux density will be zero. There is, however, a large difference in magnetic potential (AKA magneto-motive-force) between the inside and outside of the cube. The magnets can be seen as a sort of ‘magnetic battery’ that provides this difference in magnetic potential from one end of the magnet to the other, so we certainly expect to see the inside of the cube at a different magnetic potential to the outside. If there was a hole in the cube, this considerable difference in potential would drive flux through the hole, creating a ‘fringing leakage flux’ from one pole back to the other. However, as there are no such holes in this cube, or a sphere, there is simply no path for flux to return from outside to inside, and the flux density is zero. The coercive force of the NdFeB material that I have modelled is 891000 ampere-turns per meter length of magnet, in the direction of magnetization. Therefore, as these magnets are 2mm thick in their direction of magnetization, we should expect them to produce a magneto-motive-force of 891000 x 0.002 = 1782 AT. When I took this screen shot, the cursor was in the center of the cube, and at the bottom of the screen it shows that the magnetic potential at this point (or anywhere within the cube) is indeed 1782 AT. This FEA stuff really does work, and I hope some have found it of interest.
The simplest explanation of why a ‘monopole’ cannot exist is to examine one of Maxwell’s Equations, sometimes known as Gauss’s Law for magnetics, which states that the total net flux leaving a closed volume must equal zero. By definition, a magnetic monopole would have a net flux emanating from it’s surface, so a magnetic monopole would violate this Law. FEA programs make use of this Law, so it is impossible to create an FEA magnetic model of a structure that violates this law, and so it is impossible right from the outset to create a magnetic monopole with my FEA program. That said, I wanted to see what happened when I tried, so I went ahead and modelled it anyway.
However,it’s quite OK to have equal amounts of flux leaving and entering the volume, as the Law in no way says that the flux density
on any given part of a closed surface must be zero. What my FEA program actually predicts for the cube is that the flux density, B, will be zero
at all points in space, but the Law does not in itself require that to be true, for the Law would still be satisfied if the flux left the cube at some places, such as in the centre of the faces, and returned at other places such as near the corners. In the simpler case of a permanent magnet
sphere, it MUST be the case that B is zero everywhere, because symmetry dictates that there can be no place(s) on the sphere where the flux would preferentially choose to leave or enter, but you can’t use that argument for the cube.
Here is an elegant explanation of why our cube produces zero magnetic field everywhere. You will need to take my word for it that a thin, planar permanent magnet can be modelled as a current-carrying loop around the perimeter. Assume for the moment that the permanent magnet faces of the cube are ‘very thin’, so we replace each of them with a square loop of very-small-cross-section current-carrying wire, a single-turn square winding if you like. So the cube is formed by 6 such square loops of wire. Each side of the cube will therefore consist of 2 lengths of wire, one from each adjacent loop, but here is the trick. If the direction of current in each loop is chosen so that the resultant magnetic fields all point outward, then you will find (easily drawn as a sketch on a piece of paper) that on every edge, the current flows in opposite direction in the two wires, and thus no magnetic field is produced, because we all know that the net magnetic field from 2 counter-current , very thin, co-located parallel conductors is zero.
In practice, the sides of the cube are not infinitely ‘thin’ actually 2mm thick in my FEA model, but this does not change the argument. All you need to do is stack a large number of ‘very thin’ magnets together to get the thickness you want, and the field from each and every one of them still cancels, as already described. To get the sides to fit exactly you need to bevel the edges, but that does not alter the argument either, just means that each ‘thin’ square face is a slightly different dimension. Thus we see that our square ‘monopole’ cube is expected to produce zero field
everywhere, exactly as shown by the FEA, even though the Law would have permitted flux to leave the square faces in one place and return in another. This ‘square-winding-model’ also explains why if you remove one face, then the remaining 5 behave identically to the one that was removed, except of opposite sign. Try it on a sketch. It’s true, and cute as.
I find the above ‘square-winding’ argument to be rather beautiful, and it gives me great pleasure to find that the FEA model produces the same result, of zero field everywhere.
Interestingly, the same argument predicts that we can make our cube of different width, length and height, and still it will produce zero magnetic field everywhere. Anyone care to agree or disagree, or do I need to ‘stretch’ my FEA modelled cube to find out.
?
Taking it a step even further, I’ll stick my neck out and say we can build a permanent magnet ‘monopole’ of
any shape at all, and in all cases it will produce zero field at all points in space, just provided that the magnetization vector is always at right angles to the surface, and every face is the same thickness. You agree with that, John Heath?
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