We have three identical metallic spheres `A,B, and C`. `A` is given a charge `Q, and B and C` are uncharged. The following processes of touching of two spheres are carried out in succession. Each process is carried out with sufficient time. ltbgt i. A and B
ii. B and C
iii. C and A
iv. A and B
v. B and C
The final charges on the spheres are.

Options

A. `(11 Q)/(32), (5 Q)/(16),(11 Q)/(32)`
B. `(11Q)/(32),(11 Q)/(32), (5 Q)/(16)`
C. `(8 Q)/(8),(5Q)/(16),(5 Q)/(16)`
D. `(5 Q)/(16), (11 Q)/(32),(11 Q)/(32)`

Explanation

To solve the problem step by step, we will analyze the charge distribution on the spheres after each touching process.

### Step 1: Initial Charges
- Sphere A has a charge \( Q \).
- Sphere B and Sphere C are uncharged, so they have charges of \( 0 \).

### Step 2: Touching A and B
When spheres A and B are touched, the total charge is distributed equally between them since they are identical spheres.
- Initial charge on A = \( Q \)
- Initial charge on B = \( 0 \)
- Total charge = \( Q + 0 = Q \)
- After touching, the charge on each sphere:
\[ Q_{A} = Q_{B} = \frac{Q}{2} \]
- Final charges:
- Sphere A: \( \frac{Q}{2} \)
- Sphere B: \( \frac{Q}{2} \)
- Sphere C: \( 0 \)

### Step 3: Touching B and C
Now we touch spheres B and C.
- Initial charge on B = \( \frac{Q}{2} \)
- Initial charge on C = \( 0 \)
- Total charge = \( \frac{Q}{2} + 0 = \frac{Q}{2} \)
- After touching, the charge on each sphere:
\[ Q_{B} = Q_{C} = \frac{\frac{Q}{2}}{2} = \frac{Q}{4} \]
- Final charges:
- Sphere A: \( \frac{Q}{2} \)
- Sphere B: \( \frac{Q}{4} \)
- Sphere C: \( \frac{Q}{4} \)

### Step 4: Touching C and A
Next, we touch spheres C and A.
- Initial charge on C = \( \frac{Q}{4} \)
- Initial charge on A = \( \frac{Q}{2} \)
- Total charge = \( \frac{Q}{4} + \frac{Q}{2} = \frac{Q}{4} + \frac{2Q}{4} = \frac{3Q}{4} \)
- After touching, the charge on each sphere:
\[ Q_{C} = Q_{A} = \frac{\frac{3Q}{4}}{2} = \frac{3Q}{8} \]
- Final charges:
- Sphere A: \( \frac{3Q}{8} \)
- Sphere B: \( \frac{Q}{4} \)
- Sphere C: \( \frac{3Q}{8} \)

### Step 5: Touching A and B Again
Now we touch spheres A and B again.
- Initial charge on A = \( \frac{3Q}{8} \)
- Initial charge on B = \( \frac{Q}{4} \)
- Total charge = \( \frac{3Q}{8} + \frac{Q}{4} = \frac{3Q}{8} + \frac{2Q}{8} = \frac{5Q}{8} \)
- After touching, the charge on each sphere:
\[ Q_{A} = Q_{B} = \frac{\frac{5Q}{8}}{2} = \frac{5Q}{16} \]
- Final charges:
- Sphere A: \( \frac{5Q}{16} \)
- Sphere B: \( \frac{5Q}{16} \)
- Sphere C: \( \frac{3Q}{8} \)

### Step 6: Touching B and C Again
Finally, we touch spheres B and C.
- Initial charge on B = \( \frac{5Q}{16} \)
- Initial charge on C = \( \frac{3Q}{8} \)
- Total charge = \( \frac{5Q}{16} + \frac{3Q}{8} = \frac{5Q}{16} + \frac{6Q}{16} = \frac{11Q}{16} \)
- After touching, the charge on each sphere:
\[ Q_{B} = Q_{C} = \frac{\frac{11Q}{16}}{2} = \frac{11Q}{32} \]
- Final charges:
- Sphere A: \( \frac{5Q}{16} \)
- Sphere B: \( \frac{11Q}{32} \)
- Sphere C: \( \frac{11Q}{32} \)

### Final Result
- Final charges on the spheres are:
- Sphere A: \( \frac{5Q}{16} \)
- Sphere B: \( \frac{11Q}{32} \)
- Sphere C: \( \frac{11Q}{32} \)