Laplace's Equation concerning a Neutron Spin Flipper

[jmtome2]

Homework Statement

I'm taking an independent study course at my college in which my professor and I have decided to make a neutron spin flipper. To do this, I've got to solve Laplace's Equation for a cylindrical shell of (obviously) finite length and thickness. Can I assume azimuthal symmetry for the cylinder?

Homework Equations

NA

The Attempt at a Solution

I've thought about this a lot actually. It seems to me that the angle (phi) has no bearing on the solution for the potential. For example, if I rotate the shell around some arbitrary point inside the shell, it would not change the potential in the least. However, after searching for such a case in my textbook and online, I haven't come across any good examples of this. So I just want to see what you guys think. Assuming azimuthal symmetry would greatly help in the separation of variables process, but I have to make sure that I am right.

 

[mark726]

Yes, in this case, you can assume azimuthal symmetry for the cylindrical shell. This means that the potential will only depend on the radial distance from the center of the cylinder and not on the angle (phi). As you mentioned, rotating the shell around an arbitrary point inside the shell will not change the potential, further supporting the assumption of azimuthal symmetry. This assumption will make the separation of variables process much simpler and will also allow for a more straightforward solution to Laplace's equation. However, it is always important to double check your assumptions and make sure they are appropriate for the problem at hand. Good luck with your project!