Finding potential from Laplace's eq'n

  • Thread starter sachi
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We start with Laplace's eq'n in 2-d place polar co-ordinates and find all single valued separable solutions which 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) }

we also note that V is always finite

we then have to find V in certain regions with certain B.C's

we are given that V(a,theta)=Vo + Vo*cos(theta) and that V(b,theta)=2Vo*(sin^2(theta))
and we need to show that for a<r<b:
V(r,theta)=Vo{1 + a/(a^2 - b^2) * (r - (b^2)/r)*cos(theta)
- b^2/(b^4 - a^4)*(r^2 - (a^4)/r^2)*cos(2*theta)}

I can easily find the potential for r<a and r>b but am not sure what to do for middle section!! thank very much
 

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