Introduction to Electrodynamics · Problem 3.22

Potentials — David J. Griffiths

Physics · separation-of-variables-spherical

Problem

In Prob. 2.25, you found the potential on the axis of a uniformly charged disk: V(r,0)=σ2ϵ0(r2+R2−r).V(r, 0) = \frac{\sigma}{2\epsilon_0}\left(\sqrt{r^2 + R^2} - r\right). (a) Use this, together with the fact that Pl(1)=1P_l(1) = 1, to evaluate the first three terms in the expansion (Eq. 3.72) for the potential of the disk at points off the axis, assuming r>Rr > R. (b) Find the potential for r<Rr < R by the same method, using Eq. 3.66. [Note: You must break the interior region up into two hemispheres, above and below the disk. Do not assume the coefficients AlA_l are the same in both hemispheres.]

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