Question

When an uncharged conducting ball of radius R is placed in an external uniform electric field, a surface charge density σ = σ₀ cos θ is induced on the ball's surface (here σ₀ is a constant, θ is a polar angle). Find the magnitude of the resultant electric force acting on an induced charge of the same sign.

Answer 1

We know that the force acting on the area element dS of a conductor is,

It follows from symmetry considerations that the resultant force vector F directed along the z-axis, and hence it can be represented as the sum (integral) of the projection of elementary forces (1) onto the z-axis :

For simplicity let us consider an element area

Integrating this expression over the half sphere (i.e. with respect to cosθ between 1 and 0), we obtain