Two fixed point charges `+4e` and `+e` units are separated by a distance 'a'. Where should a third point charge be placed for it to be in equilibrium?

Explanation

Let a point charge q be held at a distance x from the charge +4 e, Fig. 1 (a) .20.
`:.` Distance of q from charge `+ e = (a - x)`
Force on this charge `+4` e is
`F_(1) = (q(4e))/(4pi in_(0) (a - x)^(2))` , directed away from (e)
`F_(2) = (q(e))/(4pi in_(0) x^(2))` , directed away from (e)
For this charge q to be in equilibrium `F_(1) = 0F_(2)`
i.e. `(q(4e))/(4pi in_(0) x^(2) ) = (q(e))/(4pi in_(0) (a-x)^(2))`
or `(4)/(x^(2)) = (1)/((a-x)^(2)) or (2)/(x) = (1)/(a-x)`
or `x = 2a - 2x`
`:. 3x = 2a` or `x = 2a//3`
Hence the charge q should be held at a distance `2a//3` from charge `(+4e)` e.