Classical Electrodynamics · Problem 3.7

Multipoles, Electrostatics of Macroscopic Media, Dielectrics — John David Jackson

Physics — E&M · multipole-expansion-grounded-sphere

Problem

Three point charges (q,−2q,q)(q, -2q, q) are located in a straight line with separation aa and with the middle charge (−2q)(-2q) at the origin of a grounded conducting spherical shell of radius bb, as indicated in the sketch. (a) Write down the potential of the three charges in the absence of the grounded sphere. Find the limiting form of the potential as a→0a \rightarrow 0, but the product qa2=Qqa^2 = Q remains finite. Write this latter answer in spherical coordinates. (b) The presence of the grounded sphere of radius bb alters the potential for r<br < b. The added potential can be viewed as caused by the surface-charge density induced on the inner surface at r=br = b or by image charges located at r>br > b. Use linear superposition to satisfy the boundary conditions and find the potential everywhere inside the sphere for r<ar < a and r>ar > a. Show that in the limit a→0a \rightarrow 0, Φ(r, θ, ϕ)→Q2πϵ0r3(1−r5b5)P2(cos⁡θ)\Phi(r,\,\theta,\,\phi) \rightarrow \frac{Q}{2\pi\epsilon_0 r^3}\left(1 - \frac{r^5}{b^5}\right) P_2(\cos\theta)

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