Two point charges placed a distance d apart in vacuum exert a force of magni tude F on each other. One of the two charges is doubled. To keep the magnitude of force same the separation between the charges should be changed to

The correct option is $\sqrt{2}d$.

Step 1: Understand the initial scenario using Coulomb's Law
According to Coulomb's Law, the electrostatic force $F$ between two point charges $q_1$ and $q_2$ separated by a distance $d$ in vacuum is given by the formula:
$$F=k\frac{q_1{q}_2}{d^2}$$
where $k$ is Coulomb's constant.

Step 2: Define the modified scenario
Let one of the charges be doubled. For instance, we change $q_1$ to $2q_1$ while keeping $q_2$ unchanged. Let the new separation required to keep the force magnitude at $F$ be $d^{\text{'}}$.
The expression for the force in this new scenario is:
$$F^{\text{'}}=k\frac{\left(2q_1\right)q_2}{{\left(d^{\text{'}}\right)}^2}$$

Step 3: Equate the forces and solve for the new distance
Since the problem specifies that the magnitude of the force must remain the same ($F^{\text{'}}=F$), we equate the two expressions:
$$k\frac{q_1{q}_2}{d^2}=k\frac{2q_1{q}_2}{{\left(d^{\text{'}}\right)}^2}$$
Dividing both sides by the common factors $k{q}_1{q}_2$, we get:
$$\frac{1}{d^2}=\frac{2}{{\left(d^{\text{'}}\right)}^2}$$
Taking the reciprocal of both sides:
$${\left(d^{\text{'}}\right)}^2=2d^2$$
Taking the square root on both sides to find the new separation $d^{\text{'}}$:
$$d^{\text{'}}=\sqrt{2}d$$
Therefore, the separation between the charges should be changed to $\sqrt{2}d$ to keep the force unchanged.