Three point charges q, 2q and 8q are to be placed on a
. 9cm long straight line. Find the
. positions where the charges shouldbe placed such that the potential energy
. of this sysrem is minimum. In this situation, what is the
. electric field at the charge q due to the other two charges?

Explanation

Let the charge `q` is placed at distance `x` from `2q` as shown.
`U = (1)/(4 pi in_(0)) . [ (2q.q)/(x) + (q . 8q)/(9 - x) + (2 q. 8q)/(9)]`
From `U` to be minimum , `dU//dx = 0`
`U = (q^(2))/(4 pi in_(0)) [ 2 x^(-1) + 8(9 - x)^(-1) + (16)/(9)]`
`(dU)/(dx) = 2 (-1) x^(-2) + 8 (-1) ( 9 - x)^(-2) (-1) = 0`
`(2)/(x^(2)) = (8)/((9 - x)^(2)) rArr (1)/(x) = (2)/( 9 - x) rArr x = 3 cm`
Electric field at the position of `q`
`E = (1)/( 4 pi in_(0)) [ (2q)/(x^(2)) - (8q)/((9 - x)^(2))] = (q)/( 4 pi in_(0)) [ (2)/(9) - (8)/(36)] = 0`