- #1

RickRazor

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Moved from technical forum so the template is missing

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

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

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