Two spherical conductors B and C having equal radii and cayying equal charges on them repel each other with a force F when kept apart at some distance. A third spherical conductor having same radius as that B but uncharged is brought in contact with B, then brought in contact with C and finally removed away from both. The new force of repulsion between B and C is

Options

A. `F/4`
B. `(3F)/4`
C. `F/8`
D. `(3F)/8`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Initial Setup
- We have two spherical conductors B and C, both carrying an equal charge \( Q \) and separated by a distance \( R \).
- The force of repulsion between them is given as \( F \).

### Step 2: Apply Coulomb's Law
- According to Coulomb's law, the force \( F \) between two charges is given by:
\[ F = \frac{1}{4 \pi \epsilon_0} \frac{Q_1 Q_2}{R^2} \]
- Since both charges are equal, we can write:
\[ F = \frac{1}{4 \pi \epsilon_0} \frac{Q^2}{R^2} \tag{1} \]

### Step 3: Bring the Uncharged Conductor A in Contact with B
- When the uncharged conductor A (with charge 0) comes into contact with B (with charge \( Q \)), the charges will redistribute equally because they are identical in size.
- After contact, the charge on both B and A will be:
\[ Q_A = Q_B = \frac{Q}{2} \]

### Step 4: Bring Conductor A in Contact with C
- Now, conductor A (with charge \( \frac{Q}{2} \)) is brought in contact with conductor C (with charge \( Q \)).
- The total charge when A and C are in contact is:
\[ Q_{total} = Q + \frac{Q}{2} = \frac{3Q}{2} \]
- Since they are identical, the charge will redistribute equally:
\[ Q_C = Q_A = \frac{3Q}{4} \]

### Step 5: Determine the New Charges
- After the process, we have:
- Charge on B: \( Q_B = \frac{Q}{2} \)
- Charge on C: \( Q_C = \frac{3Q}{4} \)

### Step 6: Calculate the New Force of Repulsion
- Now, we need to find the new force of repulsion \( F' \) between B and C:
\[ F' = \frac{1}{4 \pi \epsilon_0} \frac{Q_B Q_C}{R^2} \]
- Substituting the values of \( Q_B \) and \( Q_C \):
\[ F' = \frac{1}{4 \pi \epsilon_0} \frac{\left(\frac{Q}{2}\right) \left(\frac{3Q}{4}\right)}{R^2} \]
- Simplifying this gives:
\[ F' = \frac{1}{4 \pi \epsilon_0} \frac{3Q^2}{8R^2} \]

### Step 7: Relate the New Force to the Original Force
- From equation (1), we know that:
\[ F = \frac{1}{4 \pi \epsilon_0} \frac{Q^2}{R^2} \]
- Thus, we can express \( F' \) in terms of \( F \):
\[ F' = \frac{3}{8} F \]

### Final Answer
- The new force of repulsion between B and C after the process is:
\[ F' = \frac{3}{8} F \]