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My goal is to find an expression for the expression of the force on a conducting sphere due to a circular ring of total charge $Q.$ The sphere has radius $a,$ and it is placed along the axis of the ring. A vector $\mathbf{c}$ from the centre of the sphere to the radius of the ring makes an angle $\alpha$ with the axis.

To solve, first I take the origin $O$ to be the centre of the sphere and $P$ to be a point on its surface. Then the potential at a point $P$ with coordinates $r,\theta$ due to a charged ring is (Gaussian units)

$$V_1=\frac{Q}{c}\sum\limits_{n=0}^{\infty}\left(\frac{r}{c}\right)^{n}P_n(\cos\alpha)P_n(\cos\theta),~r<c.$$ Next I want to find the potential due to the induced charges, and then add the potentials together, then take the derivative to find the force. So, I would say that the potential due to the induced charges should look something like $$V_2=\frac{Q}{a}\sum\limits_{n=0}^{\infty}A_nr^nP_n(\cos\theta),$$ since it should be finite at the origin, but when applying the condition $$V(a)=V_1(a)+V_2(a)=0,$$ I am unable to find a potential which looks right, since I am unable to get the correct expression for the force, which makes me think that $V_2$ is going wrong.

Would somebody please let me know if there is an obvious issue I have overlooked? Please avoid posting full solutions; I know the correct expression for the force and I want to arrive at it myself. I've just started off by doing something dumb.

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  • $\begingroup$ Can you use the method of images? $\endgroup$ – Keith McClary Dec 5 '15 at 2:29
  • $\begingroup$ Related question? $\endgroup$ – Keith McClary Dec 5 '15 at 3:37
  • $\begingroup$ That's a related question which I saw before posting this. However I don't find it helpful because the OP doesn't ask for a method, which is what I want, and I don't know if the answer is actually correct (the expression for the force should involve a complete elliptic integral or a sum over $P_n$ and $P_{n+1}$). So I made a new topic. $\endgroup$ – user41208 Dec 5 '15 at 7:09

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