Using Legendre Polynomials in Electro

[retro10x]

Homework Statement

A conducting spherical shell of radius R is cut in half and the two halves are
infinitesimally separated (you can ignore the separation in the calculation). If the upper
hemisphere is held at potential V₀ and the lower half is grounded find the approximate
potential for the region r > R (keeping terms up to and including those ~ r^-4 ).

[Hint: you will need to expand the expression for the potential at the surface in terms of
Legendre polynomials and make use of the orthogonality of the Legendre polynomials to
determine the necessary coefficients].

Homework Equations

Multipole expansion of V; Poisson's equation with ρ(charge density)=0 when r>R

The Attempt at a Solution

So I'm having trouble starting this. My initial gut feeling would be to solve the poisson equation outside the sphere, with [itex]\nabla^{2}V(r,θ)=0[/itex] , in spherical coordinates and then apply boundary conditions.
with θ as the angle from the z-axis towards the x-y plane

However the Hint made me think that the whole system is a kind of dipole, so that I would have to use the multipole expansion. I do not understand how to expand the expression in terms of Legendre Polynomials.

I know that the solution to poisson's equation in spherical coordinates involves a legendre polynomial P_λ(cosθ) and I know the series for a legendre polynomial. The problem is that I don't know how to combine both the multipole expansion and the poisson equation solution.

Can anyone help me understand this "Hint" better, and what it means to express the multipole expansion in terms of Legendre polynomials?

 

[vanhees71]

The multipole expansion for azimuthally symmetric problems is in fact an expansion in terms of Legendre Polynomials!

 

[retro10x]

The multipole expansion for azimuthally symmetric problems is in fact an expansion in terms of Legendre Polynomials!

Unless there is another formula for the multipole expansion, but doesn't that formula only work if there is a charge density? There isn't any charge in this problem, only a fixed potential difference.

 

[retro10x]

I think I found the correct answer. I didn't use multipole expansion at all, I just used the solution to Poisson's equation in spherical coordinates. I had to use a Fourier transform to find the coefficients in terms of R, Vo and Legendre polynomials. I'm only asked for terms up to r^-4 so I only needed coefficients in the Fourier series from n=0 to n=3, so there were only 4 coefficients I had to solve for, for each one I used the corresponding Legendre term in the expression.

My solution satisfies the Boundary conditions, so thanks to the beautiful uniqueness theorem, I have my solution. :D

 

[paranormal]

The hint is suggesting that you can use the method of separation of variables to solve the Laplace equation outside the sphere. This method involves assuming a solution of the form V(r,θ) = R(r)Θ(θ) and then plugging it into the Laplace equation to get two separate equations, one for R(r) and one for Θ(θ). The solution for Θ(θ) will involve Legendre polynomials.

Once you have the solution for V(r,θ), you can use the boundary conditions at the surface of the sphere to determine the coefficients of the Legendre polynomials. This is where the orthogonality of Legendre polynomials comes in handy. You can use the integral of Pλ(cosθ)Pμ(cosθ) over the range of θ (0 to π) to determine the coefficients.

Once you have the coefficients, you can then use the multipole expansion to express the potential in terms of Legendre polynomials. This involves taking the solution you found for V(r,θ) and expanding it in a series of Legendre polynomials, with each term having a coefficient determined by the boundary conditions.

I hope this helps to clarify the hint and how to use Legendre polynomials in this problem.