A solid metallic sphere of radius `R` having charge `+ 3Q` is surrounded by a hollow spherical shell of radius `2 R` and having a charge `-Q`.
(a) Find electric field at distance `r` from centre of solid sphere `[R lt rlt 2R]`.
(b) Find potential difference between sphere and shell.
(c ) Find distribution of charge if
(i) inner sphere is earthed
(ii) inner sphere and shell are connected by a metallic wire.

Explanation

To find electeric field at `P`, assume charge inside sphere of radius `r` to be concentrated at centre `O`.
`E_(P) = (1)/(4 pi in_(0)) .(3Q)/(r^(2))`
(b) `V_(A) = (1)/(4 pi in_(0)) [ (3Q)/(R ) + (-Q)/(2R)]`
`V_(B) = (1)/(4pi in_(0)) [ (3Q)/(2R) + (-Q)/(2R)]`
`V_(A) - V_(B) = (1)/(4 pi in_(0)).(3Q)/(2R)`
(c ) (i) Let charge on inner sphere be `Q'`.
`V_(P) = (1)/(4 pi in_(0)) [(Q')/(R ) + (-Q)/(2R)] = 0`
`Q' = Q//2`

Charge distribution :

(ii) When shell is connected to sphere , potential difference between shell and sphere becomes zero.

`V_(A) gt V_(B)` , charge `3Q` from sphere flows to shell , so that `V_(A) = V_(B)`
Charge on sphere `= 0`
Charge on shell `= 2q`