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I have a hemispherical shell of radius a held at potential Vo. Using Laplace's equation, with appropriate boundary conditions, I've been able to find the potential at any point within the hemisphere as:

U(r,$\Theta$) = $\sum_{n=0}^\infty B_n (\frac ra)^n P_n[cos\Theta]$

where r is radial distance from the centre of the hemisphere
$\Theta$ is polar angle
$B_n$ is a coefficient determinable from initial conditions
$P_n$ is the legendre polynomial wrt cos$\Theta$

How would one go about calculating the potential outside of this hemisphere at any point? Could the general result above be useful?

Thanks in advance.

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1 Answer 1

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The outside must have it's own boundary conditions. If no external fields are present, then the usual boundary conditions for such problems are:

Boundedness or 0 at infinity: $$U(r=\infty)=0$$

Continuity (of the potential) at the boundary : $$U(r=a^+)=U(r=a^-)$$

The calculation steps should be about the same as inside the hemisphere.

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