The figure shows three concentric , conducting ,spherical shells `X,Y` and `Z` with radii `r, 2r` and `3r` respectively . `X` and `Z` are connected by a conducting wire and `Y` is uniformly charged to charge `Q`. Find:
(a) charges on shells `X` and `Z`,
(b) potentials of `X`and `Y`.

Explanation

(a) Let charges on `X` and `Z be q_(1)` and `q_(2)`.
By conservation of charge , `q_(1) + q_(2) = 0 rArr q_(2) = -q_(1)`
`V_(X) = (1)/(4 pi in_(0)) [ (q_(1))/(r ) + (Q)/(2r) + (q_(2))/(3r)]`
`V_(Z) = (1)/(4 pi in_(0)) [ (q_(1))/(3r) + (Q)/(3r) + (q_(2))/(3r)]`
Since `X` and `Z` are connected
`V_(X) = V_(Z)`
`q_(1) = -(Q)/(4) , q_(2) = -q_(1) = (Q)/(4)`
(b) `V_(X) = (1)/(4pi in_(0))[(q_(1))/(r ) + (Q)/(2r) + (q_(2))/(3r)]`
`= (1)/(4 pi in_(0)) [-(Q)/(4) + (Q)/(2) + (Q)/(12)] = (1)/(4 pi in_(0) r) . (Q)/(3)`
`= (Q)/(12 pi in_(0) r)`
`V_(Y) = (1)/(4 pi in_(0)) [(q_(1))/(2r) + (Q)/(2r) + (q_(2))/(3r)]`
`= (1)/(4 pi in_(0) r) [-(Q)/(8) + (Q)/(2) + (Q)/(12)] = (11 Q)/(96 pi in_(0) r)`