- #1
Bestphysics112
- 24
- 2
Homework Statement
Hello,
I'm trying to solve laplaces equation to find a solution for the potential in a pipe with the given boundary conditions:
at x=b, V=V_0
at x= -b, V = -V_0
at y=a, V=0
at y=-a, V=0
(Assume this configuration is centered on the origin, pipe as dimensions -b<=x<=b -a<=x<=a, infinite in z direction)
I have general form of solution (with exponential and trig combinations, I found this in griffith's section on separation of variables). I'm stumped on how to obtain more information from the boundary conditions. Here is what I'm referring to
https://imgur.com/a/wpKAY
For example, If the potential is 0 at y=a (which we are given), what exactly does this tell us for the coefficients? In the text, the example deals with instances in which one of the boundaries is the origin (x and y at 0), so I can eliminate a coefficient. But with the boundary at a non zero x and y coordinate, what should I do? Please don't give the full solution, but please give me a hint of sorts. Sorry if my english is poor, If my question doesn't make sense please say so and I will try to fix
Homework Equations
https://imgur.com/a/wpKAY (Potential expression obtained for cartesian coordinates.)
The Attempt at a Solution
Ok, here is my attempt at the solution. For the last two boundary conditions we are given, we can find that coefficient D=-D or C=-C. I'm not sure how to proceed. I've done similar problems where at least two of the boundary conditions lie on the origin (i.e. at x = 0 or y = 0). If it was like this, it is easy to see that a certain coefficient will be 0 and so forth. If, after plugging in y = +- a, I get the mentioned expressions, how can I determine what value C and D are? I asked someone and they said that C (or D) is zero but how can I determine which is 0? Because surely they both cannot be 0. A hint would be greatly appreciated.