A conducting sphere at potential V0 is half embedded in linear dielectric material of su...

A conducting sphere at potential V0 is half embedded in linear dielectric material of susceptibility χ_e, which occupies the region z < 0 (Fig. 4.35). Claim: the potential everywhere is exactly the same as it would have been in the absence of the dielectric! Check this claim, as follows:

(a) Write down the formula for the proposed potential V(r ), in terms of V₀, R, and r . Use it to determine the field, the polarization, the bound charge, and the free charge distribution on the sphere.

(b) Show that the resulting charge configuration would indeed produce the potential V(r ).

(c) Appeal to the uniqueness theorem in Prob. 4.38 to complete the argument.

(d) Could you solve the configurations in Fig. 4.36 with the same potential? If not, explain why.

Reference figure 4.35

Reference figure 4.36

Reference Prob. 4.38

Prove the following uniqueness theorem: A volume V contains a specified free charge distribution, and various pieces of linear dielectric material, with the susceptibility of each one given. If the potential is specified on the boundaries S of V (V = 0 at infinity would be suitable) then the potential throughout V is uniquely determined. [Hint: Integrate ∇ · (V₃D₃) over V.]