Force of attraction between two point electric charges placed at a distance `d` in a medium is `F`. What distance apart should these be kept in the same medium, so that force between them becomes `F//3`?

Options

A. `2sqrt(3)d`
B. `3d`
C. `9d`
D. `sqrt(3)d`

Explanation

To solve the problem of finding the new distance between two point electric charges such that the force of attraction becomes \( \frac{F}{3} \), we can follow these steps:

### Step 1: Understand Coulomb's Law
Coulomb's law states that the force \( F \) between two point charges \( Q_1 \) and \( Q_2 \) separated by a distance \( d \) in a medium with permittivity \( \epsilon \) is given by:

\[ F = \frac{1}{4 \pi \epsilon} \frac{Q_1 Q_2}{d^2} \]

### Step 2: Set Up the Initial Condition
Given that the force between the charges at distance \( d \) is \( F \), we can express this as:

\[ F = \frac{1}{4 \pi \epsilon} \frac{Q_1 Q_2}{d^2} \tag{1} \]

### Step 3: Set Up the New Condition
We want to find the distance \( x \) such that the force becomes \( \frac{F}{3} \). According to Coulomb's law, this can be expressed as:

\[ \frac{F}{3} = \frac{1}{4 \pi \epsilon} \frac{Q_1 Q_2}{x^2} \tag{2} \]

### Step 4: Divide the Two Equations
Now, we can divide equation (1) by equation (2):

\[ \frac{F}{\frac{F}{3}} = \frac{\frac{1}{4 \pi \epsilon} \frac{Q_1 Q_2}{d^2}}{\frac{1}{4 \pi \epsilon} \frac{Q_1 Q_2}{x^2}} \]

This simplifies to:

\[ 3 = \frac{x^2}{d^2} \]

### Step 5: Solve for \( x \)
Rearranging the equation gives us:

\[ x^2 = 3d^2 \]

Taking the square root of both sides, we find:

\[ x = \sqrt{3} d \]

### Step 6: Conclusion
Thus, the distance \( x \) that the charges should be kept apart in the same medium to achieve a force of \( \frac{F}{3} \) is:

\[ x = \sqrt{3} d \]