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## Main Question or Discussion Point

I'm interested in working out an efficient and precise numerical method to find the charge density on a conductor with an axially symmetric shape that is essentially a deformed sphere. (I have in mind something like [itex]r(\theta)=\sum \beta_\ell P_\ell(\cos\theta)[/itex], where P is a Legendre polynomial.)

My previous work on this involved putting N charges on the surface and simulating their motion until they reached an equilibrium. This is slow and doesn't give a high-precision approximation to the continuous case.

There's the relaxation method for solving the Laplace equation on a grid, but that seems to apply when the potential is known on a boundary. I think my boundary would consist of the conducting surface plus another surface far away, but I don't know a priori the potential difference between these two. The usual relaxation method would involve a 3-dimensional Cartesian lattice of points, which would not exploit the symmetry of the problem. (But I could probably fiddle with it somehow to make it effectively a 2-d grid.)

It seems like a better way to go would be to break down the surface of the grid into n coaxial circular rings, find their mutual capacitance matrix C, and then use Q=CV, where Q is a vector of the n charges and V is a vector of their n potentials. Since I know V (a vector with all its n components equal to some arbitrary value), I can find Q. The problem I'm running into here is that the mutual capacitances that are the elements of C all diverge logarithmically. This makes it all kind of a mess compared to the nice simple case of the mutual *inductance* of two coaxial loops of unequal radius, which is finite and can be found using elliptic integrals. I could do it for finite-thickness loops, but then maybe it gets messy. Possibly if I can characterize the logarithmic divergence properly, I can just renormalize it out using some approximation...? This method would otherwise be nice, because once the C matrix was found, it would be very efficient.

Another possibility would be to find the field of a loop of charge and simply simulate the flow of charge between loops until they reached equilibrium. The problem is that the field of a loop of charge is probably not easy to calculate with good numerical precision at an off-axis point.

Any suggestions?

My previous work on this involved putting N charges on the surface and simulating their motion until they reached an equilibrium. This is slow and doesn't give a high-precision approximation to the continuous case.

There's the relaxation method for solving the Laplace equation on a grid, but that seems to apply when the potential is known on a boundary. I think my boundary would consist of the conducting surface plus another surface far away, but I don't know a priori the potential difference between these two. The usual relaxation method would involve a 3-dimensional Cartesian lattice of points, which would not exploit the symmetry of the problem. (But I could probably fiddle with it somehow to make it effectively a 2-d grid.)

It seems like a better way to go would be to break down the surface of the grid into n coaxial circular rings, find their mutual capacitance matrix C, and then use Q=CV, where Q is a vector of the n charges and V is a vector of their n potentials. Since I know V (a vector with all its n components equal to some arbitrary value), I can find Q. The problem I'm running into here is that the mutual capacitances that are the elements of C all diverge logarithmically. This makes it all kind of a mess compared to the nice simple case of the mutual *inductance* of two coaxial loops of unequal radius, which is finite and can be found using elliptic integrals. I could do it for finite-thickness loops, but then maybe it gets messy. Possibly if I can characterize the logarithmic divergence properly, I can just renormalize it out using some approximation...? This method would otherwise be nice, because once the C matrix was found, it would be very efficient.

Another possibility would be to find the field of a loop of charge and simply simulate the flow of charge between loops until they reached equilibrium. The problem is that the field of a loop of charge is probably not easy to calculate with good numerical precision at an off-axis point.

Any suggestions?