Two parallel conducting plates , a distance `d` apart, are held horizontally one above the other in air . Between parallel plates , electric field `E` is uniform and directed vertically downwards . A spherical ball of radius `r`, made of a material of density `sigma` falls with constant velocity `v`. the coefficient of viscosity of air is `eta` and density of air is negligible in comparision of `sigma`. Find electric charge on ball assuming it to be positive.

Explanation

The forces acting on ball are
(a) Weight of ball `W = mg = sigma .(4)/(3) pi r^(3) g` , downward
(b) Electric force `F = qE` , downward
(c ) Viscous force `F_(v) = 6 pi eta r v` , upward
(d) Upthrust is negligible because density of air is very small.
Since ball is falling with constant velocity , net force `= 0`
`W + F = F_(v)`
`sigma .(4)/(3) r^(3) g + qE = 6 pi eta rv`
`q = (6 pi eta r v - sigma.(4)/(3) pi r^(3) g)/(E )`