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.

 

[sardar]

Hello John,

Thank you for reaching out for assistance with your problem. The surface charge density on the inside wall of a cylindrical cavity with an off-center line charge can be determined by using the concept of Gauss's law and the superposition principle.

First, let's consider the case where the line charge is located at the center of the cylindrical cavity. In this case, as you mentioned, the surface charge density is constant because all points on the inner surface of the cavity are equidistant from the line charge. This can be calculated using the equation σ = λ/2πr, where σ is the surface charge density, λ is the line charge, and r is the radius of the cavity.

Now, for the off-center case, we can use the superposition principle to determine the surface charge density. This principle states that the total electric field at a point is the vector sum of the individual electric fields from each charge present. In this case, we can consider the off-center line charge as two separate line charges, one located at the center and the other located at the off-center position. Using this concept, we can calculate the electric field at a point on the inner surface of the cavity due to the off-center line charge, and then subtract the electric field at the same point due to the line charge at the center.

Once we have the electric field at a point on the inner surface of the cavity, we can use Gauss's law to calculate the surface charge density. This law states that the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of the medium. In this case, the closed surface would be the inner surface of the cavity and the charge enclosed would be the total charge of the off-center line charge.

Using these principles, we can determine the varying surface charge density on the inside wall of the cylindrical cavity. I hope this helps and good luck with your problem.

Best,