Two point charges +e and +4e are kept at a distance 'd' units apart. Third point charge +q is placed between the two charges at a distance 'x' units from charge +e so as to be in equilibrium. The value of 'x' is

Let the charges $+e$ and $+4e$ be placed at a distance $d$ apart.

The third charge $+q$ is placed at a distance $x$ from $+e$. The distance of $+q$ from $+4e$ is $(d - x)$.

For the charge $+q$ to be in equilibrium, the net force acting on it must be zero. The force exerted by $+e$ must be equal in magnitude and opposite in direction to the force exerted by $+4e$.

$\dfrac{1}{4\pi\epsilon_0} \dfrac{e \cdot q}{x^2} = \dfrac{1}{4\pi\epsilon_0} \dfrac{4e \cdot q}{(d - x)^2}$

$\dfrac{1}{x^2} = \dfrac{4}{(d - x)^2}$

Taking the square root on both sides:

$\dfrac{1}{x} = \dfrac{2}{d - x}$

$d - x = 2x$

$3x = d$

$x = \dfrac{d}{3}$

Answer: $\dfrac{d}{3}$ units