The field distorting sphere

An earthed metallic sphere is kept in a uniform electric field E0k^ E_0 \hat{k} . If V(r,θ,ϕ)V(r, \theta ,\phi ) denotes the potential function for the region outside the sphere then the value of V(2R,π3,π2)V(3R,π3,2π3)\displaystyle \frac{V \left( 2R, \frac{\pi }{3} , \frac{ \pi }{2} \right) }{ V \left( 3R, \frac{-\pi }{3} , \frac{ 2\pi }{3} \right) } can be expressed as ab \dfrac{a}{b} for some positive coprime integers a,ba,b.

Evaluate a+ba+b.

Details and Assumptions:

  • k^\hat{k} is the unit vector along the direction of zz axis in a normal right handed Cartesian coordinate system.

  • Reference potential i.e. (V=0) (V=0) is obviously the sphere since it is grounded.

  • RR is the radius of the sphere.

Image credit: Wikipedia Bob lonescu

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