(a) Two pont charges `9 Q` and `- 25 Q` are fixed at separation `d`.At what distance from `9 Q` , electric field is zero on the line joining the charges ?
(b) A circular ring of radius `r` carries a total charge `Q` distributed uniformly over it. A small length `dl` of wire is removed . Find electric field at center due to the remaining ring.

Explanation

(a) The electric field will not be zero between the charges because at any point between `A` and `B` , direction of electric fields by both charges will be same . The electric field will be zero nearer to smaller charge , left of `A`.
`9 Q , E_(1) = (1)/( 4 pi in_(0)) . (9Q)/(x^(2))` , towards left
`-25 Q , E_(2) = (1)/( 4 pi in_(0)) .(25 Q)/((d + x)^(2))` , towards right
Since `E_(p) = 0 i.e. E_(1) = E_(2)`
`(9)/(x^(2)) = (25)/((d + x)^(2)) rArr (3)/(x) = (5)/(d + x)`
`x = (3d)/(2)`
(b) Electric field at `O` is zero due to complete ring. Charge on small length `dl`
`Q' = (Q)/( 2 pi r) dl`
When a small length of ring is removed , the parts facing each other cancel electric field but the part in front of removed portion is not balanced , hence net electrci field will be due to `Q'`.
`E_(0) = (1)/(4pi in_(0)) (Q')/(r^(2)) = (1)/( 4 pi in_(0)) . (Q dl)/(2 pi r .r^(2))`
`= (Q dl)/(8 pi^(2) in_(0) r^(3))` , towards right