Solving Spherical Harmonics Potential w/ Boundary Condition

[Lorna]

Homework Statement

Finding the potential.

Homework Equations

general solution of potential is in terms of spherical harmonics Y_l,m.

The Attempt at a Solution

My question is how do we know which l to choose given a boundry condition. for example if the potenial on the surface of a sphere of radius "a" = const*sin (theta) cos(theta). How do we know which l that corresonds to? or do we try all the l's and see which one works out?

Thanks

 

[Jhero]

for your post! To answer your question, the choice of l for the potential in terms of spherical harmonics depends on the specific boundary condition you are given. In this case, the potential on the surface of a sphere of radius "a" = const*sin (theta) cos(theta) corresponds to l = 1. This can be determined by comparing the given expression with the general solution of potential in terms of spherical harmonics, which is V(r,θ,φ) = ∑(l=0 to ∞)∑(m=-l to l) [A_l,m*r^l * Y_l,m(θ,φ)], where Y_l,m(θ,φ) are the spherical harmonics. In this case, the given expression is similar to the Y_1,1(θ,φ) term, which corresponds to l = 1. Therefore, the potential on the surface of the sphere is given by V(a,θ,φ) = A_1,1*a * Y_1,1(θ,φ). I hope this helps! If you have any further questions, please feel free to ask.