(a) The charges each `Q` are placed at `x - d , 2d , 4d ,…oo`. Find electric potential at origin `O`.
(b) A charge `+Q` is placed at each of the points `x = d , x = 3d ,x = 5d ,…..,oo` on the `x`-axis , and a charge `-Q` is placed at each of the points `x = 2d , x = 4d , x= 6d , ...,oo`.Find the electric potential at the origin `O`.

Explanation

(a) `V_(0) = (1)/(4 pi in_(0))[ (Q)/(d) + (Q)/(2d) + (Q)/(4d) + …oo]`
`= (1)/( 4 pi in_(0)) . (Q)/(d) [ 1 + (1)/(2) + (1)/(4) + …oo]`
`= (1)/(4 pi in_(0)) .(Q)/(d) [(1)/(1 - 1//2)] = (1)/( 4pi in_(0)) .(2Q)/(d)`
(b) `V_(0) = (1)/(4pi in_(0)) [ (Q)/(d) - (Q)/(2d) + (Q)/(3d) - (Q)/(4d) + ...oo]`
`= (1)/(4 pi in_(0)) (Q)/(d) [ 1 - (1)/(2) + (1)/(3) + (1)/(4) + ...oo]`
`= (1)/(4 pi in_(0)).(Q)/(d) log_(e ) 2`