A charge `Q` is uniformly distributed on a ring of radius `R`. Find the electric potential on the axis of ring at distance `x` from the centre of the ring. Sketch variation of potential with respect to `x`.

Explanation

`E = (1)/(4 pi in_(0)) .(Qx)/((R^(2) + x^(2))^((3//2))`
For `E` to be maximum, `(dE)/(dx) = 0`
`(dE)/(dx) = ((R^(2) + x^(2))^(3//2) . 1 - x .(3)/(2) (R^(2) + x^(2))^(1//2) . 2x)/((R^(2) + x^(2))^(3)) = 0`
`R^(2) + x^(2) - 3x^(2) = 0 rArr x^(2) = (R^(2))/(2)`

`x = +- (R)/(sqrt(2))`
`E_(max) = (1)/(4 pi in_(0)) .(Q.R//sqrt(2))/((R^(2) + R^(2)//2)^(3//2))`
`= (1)/(4 pi in_(0)) .(Q.R//sqrt(2))/((3R^(2))/(2) . sqrt((3)/(2)R))`
` = (Q)/(6 sqrt(3) in_(0) R^(2))`
`E` versus `x` graph :
At `x = 0 , E = 0` i.e. the graph passes through origin.
Right of `O , E` is along `+x`-axis (assumed `+ve`)
Left of `O , E` is along `-x`-axis (assumed `-ve`)
At `x = +- (R)/(sqrt(2)) ,|vec(E )|` is maximum
`x rarr oo , E rarr 0`