Magnetic Monopoles and Geometry

Take a balloon, rub it against your jumper, then stick it to a wall. Why does this work?

By rubbing it on your jumper, you’ve given it extra electrons and, since the electrons have a negative charge, the balloon now has a negative charge too. So, why does this make it stick to the uncharged wall? Because, by comparison, the wall is more positively charged – and positive and negative electric charges attract. In some materials, charged particles even can shift about a bit to give a more positive side near to the balloon, creating stronger sticking.

Charged balloon attracted to the hair of a cat. Public domain.

The balloon trick seems straightforward enough, but as is so often the case with nature, this scenario lies at the base of many deeper, unanswered questions about electricity and magnetism.

Electricity and magnetism are different manifestations of the same force: electromagnetism. A changing magnetic field will induce an electric current in a wire and vice-versa. So is there a magnetic “version” of the jumper-balloon experiment; that is, would it be possible to charge up an object magnetically?

It turns out that this very interesting question opens up a whole can of worms.

You see, scientists talk about the balloon as an electric monopole – a singly charged or polarised thing. In contrast, bar magnets are magnetic dipoles, because they have both north and south poles – one at each end.

So, since electricity and magnetism are linked, surely you could make a magnetic monopole, right? In 1931 one of the greatest twentieth century physicists, Paul Dirac, introduced this idea^[1]. But do magnetic monopoles exist, and how could you make one?

What happens if you take a bar magnet and cut it in half?

It would be reasonable to expect that cutting a bar magnet in half would leave you with separate north and south poles, but what actually happens is different - you end up with two smaller bar magnets, each with both a north and south pole! Why does this happen? We can start to explain it by going back to our balloon...

Can we create magnetic monopoles?
Image © TWDK.

Imagine a closed surface around the balloon, as if the balloon were trapped inside a beach ball. When the balloon is charged up, the electric field from it radiates from the surface like sunshine, diverging in all directions. The more you rub the balloon, the more it’s charged, or the “brighter”. We describe this radiation of electric charge as flux, the flow of the electric field. The balloon may be an electric monopole, but the net flux is zero: that is, if you added up the intensity and direction of the electric field all the way round, it would cancel out to nothing – because it’s made of lots of equal fields in opposite directions.

Let’s bring back our beach ball. Imagine an as yet impossible magnetic monopole inside it. The ball is penetrated by the flux of the magnetic field, some arrows heading in and some heading out. For our monopole, some need to be more one way than the other - they are unbalanced. Imagine the beach ball as two hemispheres, one sitting on top of the other to make the whole sphere, and the direction of one equator (if you drew an arrow on it) opposite in its direction to the other.

Wait – does this mean our magnets cancel again, deleting the monopole? Actually no: because the arrows could have different potentials, giving the monopole a finite value.

Wrecking ball at work, by Paul Goyette via Wikipedia Commons.

This is just one reason scientists think magnetic monopoles might exist. Yet none have been observed in the universe: the Monopole Problem. Luckily, having not observed monopoles is consistent with one of the mainstream theories from cosmology: the theory of rapid expansion of spacetime in the early universe. Why? Because if cosmic inflation is happening, monopoles may have been stretched out and spread out across the expanding universe.

Dirac himself thought that the very strong attractive force between magnetic monopoles might be enough to explain it - they’re just all stuck to each other^[1].

Although we still have no observational evidence for the existence of magnetic monopoles, they are by no means a fringe idea in physics, and are predicted by accepted theory. For the moment, they will just have to remain a fascinating and speculative notion...

References
why don't all references have links?
[1] Dirac, P. A. M. (1931). Quantised singularities in the electromagnetic field. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 133(821), 60-72. DOI: 10.1098/rspa.1931.0130

This post was written based on an original draft by Oliver Hillier during the final year of his Master’s degree in Theoretical Physics at Queen Mary, University of London. He is particularly passionate about the link between abstract mathematics and the real, physical world, and strongly believes that thinking about such topics should be for everyone. We are very grateful to him for his efforts.