A thin spherical shell radius of r has a charge Q 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 hemispheres, 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))/(2in_(0)) =(1)/(2 in_(0)) ((Q)/(4pi r^(2)))^(2), P_(1) = (Q^(2))/(32pi^(2) in_(0)r^(4))`
Due to -q, the electric field on the surface of the shell is `E= (1)/(4pi 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)/(4pi r^(2))) ((1)/(4pi in_(0))(q)/(r^(2)))= (Qq)/(16pi^(2) in_(0)r^(4))`
For equilibrium of the hemispherical shells `P_(2) ge P_(1)`
or `(Qq)/(16pi^(2) in_(0)r^(4)) ge (Q^(2))/(32pi^(2) in_(0)r^(4)) q ge (Q)/(2)`