Consider three identical metal spheres ` A, B` and `C`. Sphere `A` carries charge ` + 6q`, sphere `B` carries charge `-3q` and sphere `C` carries no charge . Spheres A and B are touched together and then separated. Sphere `C` is then touched to sphere `A` and separated from it . Finally the sphere `C` is touched to sphere `B` and separated from it . Find the final charge on the sphere `C`.

Explanation

To find the final charge on sphere C after the described interactions, we will follow these steps:

### Step 1: Touch spheres A and B
- Sphere A has a charge of \( +6q \) and sphere B has a charge of \( -3q \).
- When two conductive spheres are touched, they share their total charge equally.
- The total charge before touching is:
\[ Q_{total} = +6q + (-3q) = +3q \]
- Since the spheres are identical, the charge on each after they are separated will be:
\[ Q_A' = Q_B' = \frac{Q_{total}}{2} = \frac{3q}{2} = 1.5q \]

### Step 2: Touch sphere C to sphere A
- Now, sphere A has a charge of \( +1.5q \) and sphere C has a charge of \( 0 \).
- The total charge when A and C are touched is:
\[ Q_{total} = 1.5q + 0 = 1.5q \]
- After touching, the charge will again be shared equally:
\[ Q_A'' = Q_C' = \frac{Q_{total}}{2} = \frac{1.5q}{2} = 0.75q \]
- Therefore, after this step:
- Charge on sphere A: \( 0.75q \)
- Charge on sphere C: \( 0.75q \)

### Step 3: Touch sphere C to sphere B
- Now, sphere B has a charge of \( +1.5q \) and sphere C has a charge of \( +0.75q \).
- The total charge when B and C are touched is:
\[ Q_{total} = 1.5q + 0.75q = 2.25q \]
- After touching, the charge will again be shared equally:
\[ Q_B'' = Q_C'' = \frac{Q_{total}}{2} = \frac{2.25q}{2} = 1.125q \]
- Therefore, after this step:
- Charge on sphere B: \( 1.125q \)
- Charge on sphere C: \( 1.125q \)

### Final Result
The final charge on sphere C is:
\[ \text{Final charge on sphere C} = 1.125q \]