Finding potential from Laplace's eq'n

[sachi]

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

 

[mighty2000]

you are familiar with the process of solving differential equations and finding solutions that satisfy certain boundary conditions. In this case, the general solution for Laplace's equation in 2D polar coordinates is given as:

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) }

From the given boundary conditions, we can determine the values of A, B, Cn, Dn, En, and Fn.

First, at r=a, we have:

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

This gives us two conditions: A+Blna = Vo and Cn*En*a^-n + Dn*Fn*a^n = 0.

Similarly, at r=b, we have:

V(b,theta) = A + Blnb + sum from 1 to infinity of {(Cn*sin(n*theta) + Dn * cos(n*theta))*(En*b^-n + Fn *b^n) } = 2Vo*sin^2(theta)

This gives us two conditions: A+Blnb = 0 and Cn*En*b^-n + Dn*Fn*b^n = 2Vo*sin^2(theta).

From these four conditions, we can solve for the coefficients A, B, Cn, Dn, En, and Fn.

Next, we need to find the potential for the region a<r<b. We can do this by using the general solution and substituting the values of A, B, Cn, Dn, En, and Fn that we have solved for.

So, for a<r<b, the potential is given as:

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

= Vo + Vo*cos(theta) + sum from 1 to infinity of {[(Cn*En*a^-n + Dn*Fn*a^n) * (b^-n - a^-n)/(-