Find the electrostatic force of interaction between two halves of a spherical conductor of radius R carrying a charge Q.

Explanation

Take an elementary area dA containing a charge dq on
the elementary ring of radius r. The force acting on dq is
`dF = (sigma^2)/(2epsilon_0)dA` (as derived earlier)
The net force on the ring is
`F_(ring) = int dF sin theta = int (sigma^2)/(2epsilon_0)dA sin theta`
`= sigma^2/(2epsilon_0)sin theta intdA = sigma^2/(2epsilon_0)sin theta (A_(ring))`
where `A_(ring) = 2pir dtheta = 2pi(Rcostheta)R d theta = 2pi R^2 cos theta dtheta`
or `F_(ring) = (sigma^2piR^2)/(epsilon_0)` sin theta cos theta dtheta` ...........(i)
`F_(hemisphere) = int F_(ring)` ..........(ii)
Substituting `F_(ring)` from Eq. (ii), we get
`F_(hemisphere) = (sigma^2piR^2)/epsilon_0` in`t_(0)^(+pi//2)` sin thetacostheta, where sigma `= Q/(4piR^2)`
`= (Q^2)/(32piepsilon_0R)`