How to Solve Laplace's Equation for a Point Charge Near a Grounded Plane?

[RickRazor]

A point charge q is situated at a distance d from a grounded conducting plane of infinite extent. Find the potential at different points in space.

I want to solve this problem without using the image charge idea.

I assumed azimuthal symmetry and took the zonal harmonics. And we know that as r tends to infinite the potential in the opposite side of the conducting plane goes to a constant. So we are left with the Pn(theta)r^{-(n+1)} terms.

$$V(r,\theta)=A_1 + \frac{C_1}{r} + \frac{C_2}{r^2}\cos(\theta)) + \frac{C_3}{2}(3\cos^2(\theta) - 1) + ...$$

When cos(theta)=d/r, the potential vanishes. So we get the condition:

$$A_1 + \frac{C_1}{r} + \frac{C_2}{r^2}\frac{d}{r} + \frac{C_3}{2}(3\frac{d^2}{r^2} - 1) + ... =0$$

for all r

Now how to progress further? How do I get the coefficients

 

[jasonRF]

I don't think you are answering the same question in the image - it just asks for the total charge on the plane (which you can compute from the image solution). What is the real question you are trying to answer?

Jason

 

[jasonRF]

By the way, if you want to solve without images, you will have better luck I think if you use Cartesian or cylindrical coordinates since your boundary is a plane. It will be fairly involved - more difficult than the kinds of problems in a typical undergrad (in US) textbook like Griffiths. For fun I solved it using integral transform techniques and contour integration, but other approaches could be used. I'm pretty sure that whatever way you solve it, instead of a discrete sum of functions you will find a continuous sum (an integral). EDIT: the integral can be evaluated to yield the image solution, of course.

Jason