A conducting sphere of radius R at potential zero is surrounded by a concentric

Question

A conducting sphere of radius R at potential zero is surrounded by a concentric spherical shell of dielectric material of inner radius R, outer radius 2R, and dielectric constant ?=7/5. Suppose that the sphere and the dielectric shell are placed in an initially uniform electric field E0 in the z direction.

Show that the potential is given by:

?(r,?)=-E0rcos? + (2E0R3/r2 )cos? for r?2R

?(r,?)=-(6/7)E0(r - R3/r2)cos? for R?r?2R

I have this problem mostly solved, but I still need to know where the 2 in the first equation comes from.

It should be (?2

Explanation / Answer

You can use Gauss's law to determine the electric field outside both conductors to be q_in/(4pr²e) = (-3 nC + 2 nC)/(4pr²e) = -10^(-9) C /(4pr²e) for r > 9. For 6

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