Opposite charges repel, don’t they?

Well, the answer to that is, um… well…. it depends….

Now, I’m not suggesting what you’ve learned at school is not true. Take a point charge (e.g. a proton), and bring it close to another point charge (e.g. another proton) and the two will repel, with an inverse square law  (Let’s not take them close enough to exhibit nuclear forces). If we double the distance between the charges, the force of repulsion quarters. That’s Coulombs Law.

Things get rather more complicated, however, if the charge is not point-like. Yesterday, at the NZ Institute of Physics conference, John Lekner gave a fascinating account of the force between two charged, conducting spheres. It is actually intensely complicated and far from obvious. This problem was tackled in the nineteenth century by two physics geniuses: James Clerk Maxwell and William Thomson (Lord Kelvin). Their work was all the more remarkable given that they didn’t have symbolic algebra computer packages available to them to handle the rather intensive algebra generated by this problem. The problem is now being revisited because it has application in nanotechnology, where the conducting spheres in question are really, really small.

So, in broad terms, what happens? Why is it more complicated than Coulomb’s law? To see this, look at the case of two conducting spheres in close proximity, but one with a charge (say positive) and one neutral. The two attract!  The first sphere, with the charge, creates an electric field that is felt by the second sphere. The electrons in the second sphere will be attracted towards the first one, since the electrons are mobile in a conductor, and therefore the second sphere will polarize – its negative charge will move towards the part of the sphere that is nearest the first one, giving a separation in the positive and negative charges in the second sphere. The net force between the two spheres is then no longer zero, because the attractive force between the positive charge in the first sphere and the (nearby)  negative on the second is greater than the repulsive force between the positive charge in the first and the (further away) positive on the second.

So, a charged sphere and a neutral sphere attract. Let’s now make the second sphere slightly positive as well. If we only make it a little bit positive, the attractive force due to the polarization still wins out over the extra repulsive force due to more positive charge, and so the two still attract each other! Make it too positive, however, and the Coulomb interaction wins out and the two will repel, as we might expect.

It’s actually (much) more complicated than this, since the separation of charge in the second sphere sets up its own field that causes a separation in the first sphere, which influences the second, and so forth, creating, in mathematical terms, an infinite series of interaction terms. It’s this series that Maxwell and Kelvin grappled with, with surprising success.

So, in summary, we have a problem that seems pretty simple: "I take a conducting sphere of radius a, charged with a charge q1, place it a distance x away from a second conducting sphere of radius b charged with a charge q2. What is the force between them?" But in practice the solution is really, really nasty and not at all obvious. 

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