Two equal point charges are fixed at `x=-a` and `x=+a` on the x-axis. Another point charge Q is placed at the origin. The change in the electrical potential energy of Q, when it is displaced by a small distance x along the x-axis, is approximately proportional to

Explanation

`U_(1) = (1)/(4 pi in_(0)) (Q q)/(a) xx 2`
`= ( 2Qq)/(4 pi in_(0) a)`
`U_(2) = (1)/(4 pi in_(0)) [(Qq)/(a + x) + (Qq)/(a - x)]`
` = (Qq)/(4pi in_(0)) (2a)/((a^(2) - x^(2)))`
`= (Qq)/(4 pi in_(0)) . (2 a)/(a^(2) (1 - (x^(2))/(a^(2)))) = (2 Qq)/(4 pi in_(0) a) (1 - (x^(2))/(a^(2)))^(-1)`
`= (2Qq)/(4 pi in_(0)) (1 + (x^(2))/(a^(2)))` [ since `x lt lt a`]
`Delta U = U_(2) - U_(1) = (2 Qq)/(4 pi in_(0) a) .(x^(2))/(a^(2)) = (1)/(4 pi in_(0)) .(2 Qqx^(2))/(a^(3))`