Conducting Sphere in a nonconducting fluid

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Homework Statement

I had a question there is a conducting sphere submerged in a nonconducting fluid with a dielectric constant do I treat the fluid as if it is a uniform field? Or do I go ahead and use Legendre polynomials? I asked to find E,D and the polarization.

Homework Equations

The Attempt at a Solution

 

[gabbagabbahey]

Assuming your dielectric fluid is linear and fills all of space, the fields (E and D) in the fluid will take on the symmetries of the electric field due to the conductor.