For some work I am doing I wish to be able to define the potential distribution as a function of the radius (ρ) between two concentric electrodes.
One solution (from reliable literature) defines the varying radial potential as:
Where V0 and ρ0 are the potential and radius of the optic axis respectively.
The Attempt at a Solution
Taking the laplace equation in spherical coordinates I find the potential is given by:
This assumes there is no variance in potential in θ or [itex]\varphi[/itex] which I'm confident is correct providing there is a uniform surface charge distribution across both electrodes. However the steps after this become pretty complex involving Legendre polynomials. I'm certainly no where near deriving the above equation.
Any help or suggestions anyone may have would be very welcome!