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Homework Statement
First of all can i say that my question is part of a bigger problem that I'm trying to solve, but for the moment I'm stuck at this bit!
I'm trying to obtain a function that will return the temperature at a chosen point [itex](r,\theta)[/itex] on a disk of radius [itex]r_{0}[/itex]. The temperature of the disk is constant across its area.
Homework Equations
So far what I have done is use a general solution for the polar form of the 2d steady state laplace equation;
[itex]f(r,\theta) = (1/2)a_0 + \sum\limits_{n} (r/r_0)^n (a_n cos(n\theta) + b_n sin(n\theta))[/itex]
and solved for the Fourier coefficients using [itex]g(\theta) = T[/itex] where [itex]T[/itex] is the temperature distribution around the disk's perimeter.
The Attempt at a Solution
I had hoped to get an equation that would give the same temperature for all points on the disk but instead i have this;
[itex]f(r,\theta) = T + \sum\limits_{n} (r/r_0)^n (T/\pi) cos(n\theta)[/itex]
which returns varying temperatures depending on which point i choose. some points are warmer than the disk itself! something is obviously wrong. i believe that I'm thinking about the problem the wrong way or I've not fully understood what I'm trying to do. the idea is that i can eventually choose a point outside the disk and find the temperature there, then plot a heat flow graph with other boundaries at different temperatures.