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I have a sphere where the upper half surface has a potential of $V_0$ and the lower half is grounded and I have to find the potential everywhere (using the Laplace solution for spherical coordinates). But when I try to equate $\Phi(R,0)=V_0$ (for $r=R$ on the surface and $\theta=0$ for the top of the sphere) and $\Phi(R,\pi)=0$ for the lower end, nothing much comes out for the $B$ terms of the multipole.

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    $\begingroup$ What do you mean by the 'B terms'? $\endgroup$ – Gert Sep 6 '20 at 18:51
  • $\begingroup$ From the Laplace solution for spheres.There are Bl/r^(l+1) terms and Al*r^l and I've only kept the B ones since I don't want to solution to tend to infinity for r that tends to infinity. $\endgroup$ – Marianna Sep 6 '20 at 19:05
  • $\begingroup$ “A terms” and “B terms” are not standard terminology. Different people use different symbols for these coefficients (although $A$ and $B$ are indeed common). $\endgroup$ – G. Smith Sep 6 '20 at 19:34
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You need to keep $r^l$ terms for $0 \le r \le R$ and $r^{-l-1}$ terms for $R \le r < \infty$.

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