A charge $Q$ is divided into two parts and placed at a given distance so that the force between them is maximum. The value of each charge is:

Let the two parts of the charge $Q$ be $q$ and $Q - q$.

The electrostatic force between them when placed at a given distance $r$ is given by Coulomb's law:

$F = \dfrac{1}{4\pi\varepsilon_0} \dfrac{q(Q - q)}{r^2}$

For the force to be maximum, the first derivative of $F$ with respect to $q$ must be zero:

$\dfrac{dF}{dq} = 0$

$\dfrac{1}{4\pi\varepsilon_0 r^2} \dfrac{d}{dq}(qQ - q^2) = 0$

Since $\dfrac{1}{4\pi\varepsilon_0 r^2}$ is a non-zero constant, we get:

$\dfrac{d}{dq}(qQ - q^2) = 0$

$Q - 2q = 0$

$2q = Q$

$q = \dfrac{Q}{2}$

The other part of the charge is $Q - q = Q - \dfrac{Q}{2} = \dfrac{Q}{2}$.

Thus, the value of each charge is $\dfrac{Q}{2}$ and $\dfrac{Q}{2}$.