Three identical conducting spheres A, B and C are given charges +Q, +q and -2Q respectively. A and C are brought in to contact and separated. Next B and C are brought in to contact and separated. If A,B and C are arranged so that their centres form an equilateral triangle and they do not touch each wihich of the following statement is correct regarding the forces between the pairs
(A,B),(B,C), and (C,A)?

Options

A. (A,B) atrract, (B,C) attract, (C,A) repel
B. (A,B) attract, (B,C) repel, (C,A) repel
C. (A,B) atrract, (B,C) repel, (C,A) attract
D. (A,B) repel, (B,c) attract, (C,A) repel

Explanation

To solve the problem step by step, we will analyze the charges on the spheres after they are brought into contact and then determine the forces between them.

### Step 1: Initial Charges
- Sphere A has a charge of \( +Q \).
- Sphere B has a charge of \( +q \).
- Sphere C has a charge of \( -2Q \).

### Step 2: Contacting Spheres A and C
When spheres A and C are brought into contact, the total charge will be distributed equally between them because they are identical conducting spheres.

- Total charge when A and C are in contact:
\[ Q_A + Q_C = +Q - 2Q = -Q \]
- Each sphere will have half of this total charge after separation:
\[ Q_A' = Q_C' = \frac{-Q}{2} \]

### Step 3: New Charges on A and C
After separation:
- Sphere A now has a charge of \( -\frac{Q}{2} \).
- Sphere C also has a charge of \( -\frac{Q}{2} \).

### Step 4: Contacting Spheres B and C
Next, we bring sphere B (which has a charge of \( +q \)) into contact with sphere C (which now has a charge of \( -\frac{Q}{2} \)).

- Total charge when B and C are in contact:
\[ Q_B + Q_C' = +q - \frac{Q}{2} \]
- Each sphere will again have half of this total charge after separation:
\[ Q_B' = Q_C'' = \frac{+q - \frac{Q}{2}}{2} \]

### Step 5: New Charges on B and C
After separation:
- Sphere B now has a charge of \( \frac{+q - \frac{Q}{2}}{2} \).
- Sphere C now has a charge of \( \frac{+q - \frac{Q}{2}}{2} \).

### Step 6: Final Charges
- Sphere A: \( -\frac{Q}{2} \)
- Sphere B: \( \frac{+q - \frac{Q}{2}}{2} \)
- Sphere C: \( \frac{+q - \frac{Q}{2}}{2} \)

### Step 7: Analyzing Forces Between Spheres
Now, we need to analyze the forces between the pairs of spheres:
1. **Force between A and B**:
- A has a negative charge \( -\frac{Q}{2} \) and B has a positive charge \( \frac{+q - \frac{Q}{2}}{2} \).
- Since opposite charges attract, there will be an attractive force between A and B.

2. **Force between B and C**:
- Both B and C have the same charge \( \frac{+q - \frac{Q}{2}}{2} \).
- Since they are both positive, they will repel each other.

3. **Force between C and A**:
- C has a charge of \( \frac{+q - \frac{Q}{2}}{2} \) and A has a charge of \( -\frac{Q}{2} \).
- Again, since opposite charges attract, there will be an attractive force between C and A.

### Conclusion
- The forces between the pairs are:
- A and B: Attractive
- B and C: Repulsive
- C and A: Attractive

Thus, the correct statement regarding the forces between the pairs (A, B), (B, C), and (C, A) is:
- A and B attract each other, B and C repel each other, and C and A attract each other.