We have laplace's eqn in 2-d polar co-ordinates.(adsbygoogle = window.adsbygoogle || []).push({});

We find separable solutions where V(theta) = V(theta + 2Pi)

This gives us the general solution

V(r,theta) = A + Blnr + sum from 1 to infinity of {(Cn*sin(n*theta) + Dn * cos(n*theta))*(En*r^-n + Fn *r^n) }

where A,B,C,D,E,F are all constants

we are given the boundary conditions V tends to zero as r tends to zero, therefore we conclude that A=B=Dn=0 therefore we end up with

V = sum of {(Cn*sin(n*theta) + Dn*cos(n*theta))*r^-n}

We are given further B.C's that at r= r0

V= 2*Vo*theta/Pi for -Pi/2<theta<=Pi/2

V=2*Vo*(1-theta/Pi) for -Pi/2<theta<=Pi/2

N.b Vo is a constant

at this point we clearly need to take an F.S but I'm a bit confused as the function has so symmetry about theta=0 and this messes up the algebra a lot.

I think the solution might be to translate the theta function by Pi/2 in the -ve theta direction to make it even, and then take the Fourier cosine series, then using the substitution theta = theta + Pi/2 to turn in back into the original function. Is this is appropriate method, or is there a different way? Thanks very much for your help.

Sachi

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Laplace's eqn solution

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**