A thin spherical shell radius of r has a charge 2 uniformly distributed on it. At the centre of the shell, a negative point charge -q is placed. If the shell is cut into two identical hemi spheres, still equilibrium is maintained. Then find the relation between Q and q?
 

Explanation

Here the outward electric pressure at every point on the shell due to its own charge is
`P_1 = (sigma^2)/(2 in_0) ((Q)/(4pi r^2))^2 , P_1 = (Q^2)/(32 pi^2 in_0 r^4)`
Due to -q, the electric field on the surface of the shell is `E = (1/(4 pi in_0) q/(r^2))`. This electric field pulls every point of the shell in inward direction. The inward pressure on the surface of the shell due to the negative charge is `P_2 = sigma E`
`= ((Q)/(4 pi r^2)) (1/(4 pi in_0) q/(r^2)) = (Qq)/(16 pi^2 in_0 r^4)`
For equilibrium of the hemispheircal shells `P_2 ge P_1`
or `(Qq)/(16 pi in_0 r^4) ge (Q^2)/(32 pi^2 in_0 r^4) "     " q ge (Q)/(2)`.