When an uncharged conducting ball of radius `R` is placed in an external uniform electric field, a surface charge densityh `sigma = sigma_(0) cos theta` is induced on the ball's surface charge (here `sigma_(0)` is a constant, `theta` is a polar angle). Find the magnitude of the resultant electric force acting on an induced charge of the same sign.

Explanation

We know that the force acting on the area element `dS` of a conductor is,
`d vec(F) = (1)/(2) sigma vec(E) dS` …(1)
It follows from symmetry considerations that the resultant force `vec(F)` is directed along the z-axis, and hence it can represented as the sum (intergal) of the projection of elementary forces (i) onto the z-axis :
`dF_(x) = dF cos theta` ...(2)
For simplicity let us consider an element area `dS = 2pi R sin theta R d theta` (Fig). Now considering that `E = sigma//epsilon_(0)`. Equaction (2) takes teh form
`dF_(x) = (pi sigma^(2) R^(2))/(epsilon_(0)) sin theta cos theta d theta`
`= - ((pi sigma_(0)^(2) R^(2))/(epsilon_(0))) cos^(2) theta d cos theta`
Intergating this expression over the half sphere (i.e with respect to `cos theta` between 1 and 0), we obtain `F = F_(x) = (pi sigma_(0)^(2) R^(2))/(4 epsilon_(0))`