Numerical method for finding charge density on a conductor

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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?
 

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bcrowell
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It turns out that you can get the potential of a charged ring in terms of an elliptic integral. The following is from online lecture notes by Vittorio Celli:
First we review the similar problem of computing the electrostatic potential
of a uniformly charged ring, which is

[tex]

\Phi (\rho ,z)=\frac Q\pi \int_0^\pi \frac{d\phi ^{\prime }}{\sqrt{%

r^2+a^2-2a\rho \cos \phi ^{\prime }}}

[/tex]
The integral is transformed to standard form by the change of variable
[itex]\phi^{\prime }=\pi -2\varphi [/itex], [itex]\cos \phi ^{\prime }=2\sin ^2\varphi -1,[/itex] so that

[tex]
\sqrt{r^2+a^2-2a\rho \cos \phi }=\sqrt{a^2+r^2+2a\rho }\sqrt{1-\frac{4a\rho
}{a^2+r^2+2a\rho }\sin ^2\varphi }
[/tex]

The result is

[tex]

\Phi =\frac{2Q}{\pi \sqrt{a^2+r^2+2a\rho }}K\left( \frac{4a\rho }{%

a^2+r^2+2a\rho }\right)

[/tex]

where [itex]K(m)[/itex] is the complete elliptic integral of the first kind


[tex]K(m)=\int_0^{\pi /2}\frac{d\varphi }{\sqrt{1-x\sin ^2\varphi }} [/tex]
There are also other ways of expressing this, such as with a series involving Legendre polynomials: http://leofangyaolung.files.wordpress.com/2012/05/potential1.pdf [Broken] (discusses both the disk and the ring). This approach is also in Jackson.

This would seem to make it pretty straightforward to calculate the mutual capacitances, although there would still be the issue of the logarithmic divergences.
 
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Electrostatics is strictly similar to gravitation, and algebraic solutions exist for the gravitation field of a flattened sphere; the field is expressed as a sum of (Bessel?) special functions.

This is needed to find the orbit of satellites around our Earth, especially low-orbiting (800km) satellites with inclination nearly polar like 98°, where the equatorial bulge lets the orbit's plane rotate by 1 turn in 1 year, so the satellite passes always at the same Solar time over the ground.
 
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Electrostatics is strictly similar to gravitation, and algebraic solutions exist for the gravitation field of a flattened sphere; the field is expressed as a sum of (Bessel?) special functions.
Cool stuff -- both the math (which I wouldn't have thought was doable in closed form) and the application. But I don't think it directly helps me, since my shapes aren't necessarily ellipsoids, and the charge density (which is the unknown I'm trying to find) is nonuniform. The charge density happens to be known in closed form for a conducting ellipsoid: http://math.stackexchange.com/questions/112662/gaussian-curvature-of-an-ellipsoid-proportional-to-fourth-power-of-the-distance My motivation for studying this numerically is to see if there is a strict relationship between curvature and charge density, as there is for an ellipsoid and for certain other special shapes: https://www.physicsforums.com/showthread.php?p=4047030
 
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It sounds like you might be able to solve that nicely using Conformal Mapping: http://mathworld.wolfram.com/ConformalMapping.html

I have no experience with the technique but was thinking of using it to solve a similar type of problem years ago. It requires you to cast the problem into a representation using complex numbers. There are a couple of books to recommend (which I have) but at the moment I don't have the references.
 
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It sounds like you might be able to solve that nicely using Conformal Mapping: http://mathworld.wolfram.com/ConformalMapping.html

I have no experience with the technique but was thinking of using it to solve a similar type of problem years ago. It requires you to cast the problem into a representation using complex numbers. There are a couple of books to recommend (which I have) but at the moment I don't have the references.
Yeah, unfortunately I think that only works for a problem that reduces to a plane. My problem is effectively two-dimensional because it has cylindrical symmetry, but it doesn't reduce to a plane.

I think I have a strategy that works. Since we have a closed-form result for the potential of a ring, it becomes trivial to compute the elastance matrix (inverse of the capacitance matrix), P. The diagonal elements of this matrix are infinite if the rings have zero thickness, but we can treat them as having a thickness equal to the separation a between neighboring rings. The diagonal elements then become something more like (I think) [itex](k/r)\ln(r/a)[/itex], where r is the radius of that particular ring.

Once I have the elastance matrix, I can put some charge on each ring, and then in O(n^2) time I can find the potential difference between neighboring rings. Taking some finite resistance between neighbors, I can simulate a current, and iterate until I get to equilibrium.
 
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