# Electric potential for half-grounded sphere [closed]

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.

• What do you mean by the 'B terms'?
– Gert
Sep 6 '20 at 18:51
• 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. Sep 6 '20 at 19:05
• “A terms” and “B terms” are not standard terminology. Different people use different symbols for these coefficients (although $A$ and $B$ are indeed common). Sep 6 '20 at 19:34

You need to keep $$r^l$$ terms for $$0 \le r \le R$$ and $$r^{-l-1}$$ terms for $$R \le r < \infty$$.