Surface charge density on a cylindrical cavity

[rockbreaker]

Hi folks, I am having trouble generalizing a well-known problem. Let's say we have a cylindrical cavity inside a conductor, and in this cavity runs a line charge λ. I would now like to know the surface charge density on the inside wall of the cavity, but with the line charge not in the center of the cylindrical cavity.

It's clear that if the line charge is located in the center, the surface charge density is a constant because all points of the inner surface of the cavity are equally close to the line charge.

So when the line charge is off-center, the surface charge distribution has to be varying around the center with the angle. Yet, the inner surface of the cavity still has to be a equipotential surface.

Can anyone help me with an idea of how to solve this problem? I will for sure need the cosine law to determine the distance of the surface of the cavity from the line charge, but from there...?

Thank you very much for your help!

Regards, John

 

[Meir Achuz]

Write the potential as a Fourier cosine series in a_n r^n cos\theta
plus ln[\sqrt{r^2+d^2-2rd cos\theta}].
Expand the log in a Fourier cosine series . Then set the potential = 0 at the surface r=R, setting each term in the series to zero to find the coefficients a_n.

 

[rockbreaker]

That's a great idea, thank you very much. I first tried to solve this problem with image charges, but the problem is that I don't know where to place it. In the solution of Laplace's equation, all coeficients for any term r^n for n<0 must be zero, but can there survive any others than the logarithmic term?

(Two line charges a distance L/2 apart produce a potential λ/2∏ε0*ln((r^2+(L/2)^2-rLcosθ)/(r^2+(L/2)^2+rLcosθ)).)

 

[Meir Achuz]

Inside the cylinder, n must be zero or positive, with non negative and no log.