- #1
skrat
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Homework Statement
A long pipe is split to four four pieces along its length. The quarters are slightly moved apart and attached to constant electric potentials ##U_0## and ##-U_0## as shown in the cross section of the pipe in the attached picture.
The walls of the pipe are thin and the distance between the quarters is small compared to radius ##a## of the pipe.
Find the electric potential inside the pipe as function of polar coordinates ##r## and ##\varphi ## and parameters ##U_0## and ##a##. You can leave the result in infinite series.
Homework Equations
##U(r,\varphi )=A_0+B_0ln(r)+\sum_{n=0}^{\infty}\left [ (A_mr^m+B_mr^{-m})\cos(m\varphi)+(C_mr^m+D_mr^{-m})\sin(m\varphi) \right ]##
The Attempt at a Solution
Now I know that all parts EXCEPT ##C_mr^m \sin( \varphi m)## should vanish in the series written in relevant equations but the problem is that I don't really understand how this happens.
Here is how I tried:
- Of course, we don't want the potential inside the pipe to have any infinite values at all, therefore as ##r->0## it is obvious that ##B_0=B_m=D_m=0##. This leaves me with $$U(r,\varphi )=A_0+\sum_{n=0}^{\infty}\left [ A_mr^m\cos(m\varphi)+C_mr^m\sin(m\varphi) \right ]$$
And now I have no idea on how to continue. I have no clue what other boundary conditions should be in order to get something useful.