Find the potential - Long hollow cylinder

[DieCommie]

Homework Statement

Consider a long hollow cylinder of radius b that is divided into equal quarters with alternate segments being held at potential $$+V_0$$ and $$-V_0$$.

a) Find the potential inside the cylinder.

b) Sketch the equipotentials.

Homework Equations

Is this an application of Laplace's equation?

The Attempt at a Solution

Dont know where to start!


Thx.

 

[Gokul43201]

Yes, it is an application of Laplace's equation. Do you know the solution of the LE in cylindrical co-ordinates? Have you seen Bessel functions before? If not, you need to first study this. You will find it covered in most any standard E&M text (eg: Jackson, and probably Griffiths too).

 

[DieCommie]

Yes I have seen those before, but am not very good at them. From digging around texts I come to the conclusion that the solutions are modified bessel functions. This is because of the cylindrical shape. Does this seem correct?

Now I am running into problems applying the boundary conditions. The potential goes from $$+V_0$$ to $$-V_0$$ at $$\phi = 0$$. Does that mean the potential at $$\phi = 0$$ is $$0$$? (And likewise at $$\phi = \frac{\pi}{2}, \pi, \frac{3\pi}{2}$$?) When many of my constants go to zero and I lose my $$\phi$$ dependence. I know this is not correct because the potential must be dependent on $$\phi$$...

Very confusing for me! and I am sure its confusing for you trying to help me, but if you have any suggestions (especially pertaining to boundary conditions) please tell! Thank you.

 

[Gokul43201]

Yes, I'm confused. What are theta and phi? In cylindrical co-ordinates you have only one of them.

Here's the general approach:

Write down the general solution of the LE in cylindrical co-ordinates. Look at the radial part - you can eliminate one of the coefficients by inspection. Then look at the angular part. Again, you can eleiminate one coefficient by noticing the symmetry of the boundary potential (for a particular choice of axes, it will be either even or odd). Now finally, set the potential equal to its boundary values at the boundaries and use this to evaluate the Fourier and Bessel coefficients.

 

[DieCommie]

Sorry about that. I change them all to $$\phi$$. I am going to look it over one last time before I ask any other questions. Thanks for your time so far!