The field distorting sphere

Electricity and Magnetism Level 4

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

Evaluate \(a+b\).

Details and Assumptions:

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

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

  • \(R\) is the radius of the sphere.

Image credit: Wikipedia Bob lonescu

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