- #1

Diane Wilbor

- 7

- 1

## Homework Statement

A grounded Z-axis symmetric closed conductor has a single point charge at the origin within it, inducing negative charge onto its inner surface.

Given the induced charge density from the unit point charge, find the surface charge induced instead by a unit dipole at the origin, oriented in the +X direction.

## Homework Equations

I'm unsure!

## The Attempt at a Solution

I can write down the answer by inspection. If the surface density from the point charge has some (given!) $$\sigma(r, z, \theta) = f(r, z)$$ then the dipole induced density will influence only the ##\theta## dependence, with the obvious cosine dipole weighting, and thus the answer will be proportional to $$\sigma_{\rm dipole}(r, z, \theta)=f(r, z)\cos\theta.$$ As a check for this intuition, it does hold true for a dipole inducted charge for a conductive sphere. Spherical coordinates would be the same, but cylindrical coordinates seem more natural for this kind of problem.

The constant of proportionality is unknown, but we can integrate our (proposed) solution and find that we need to divide by the mean radius, ##\int f(r,z)\ r\ dr\ dz.##

But the hard part is

*proving*this. I first thought to expand potential in terms of a multipole expansion of the potential in cylindrical coordinate, then seeing which terms get "shifted" with a cosine weighting of ##\theta## but this doesn't go anywhere since the multipole expansion is just 0 (for the system as a whole) or a unit radial (for the conductor's contribution to the farfield expansion.)

Another interesting thought is that if the system were spherically inverted with a Kelvin transformation, we'd have the same problem but with an initial system of an axial symmetric charged capacitor and relating that surface density to the surface charge induced into the same conductor in a linear external field. Interesting, but it doesn't give any new tool to analyze the problem.

The problem specifies the conductor is closed. I suspect this is an unnecessary condition and the problem is valid for any axially symmetric geometry, closed or open, and with no need for the origin to be within the conductor's interior. Again this gives no help to the answer, but was just another observation when thinking about the problem.

I considered a simple substitution. If we replace the central charge with a +X dipole, we would still be in balance if every bit of surface charge was replaced with a proporionally scaled +X dipole as well. Those surface dipoles are not normal to the conductor. But how to transform the oriented surface dipole into an equivlent charge? The ##\cos \theta## term could come from ##{p}\ \cdot\ {\hat{n}}## but what's the justification for that?

I'm running out of tools to attack the problem. This is for self-study, so I am eager to gain more intuition about how to think about such manipulations. Thanks for any help!