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## 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 V

_{0}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].

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?