Three isolated metal spheres A, B and C have radii R, 2R and 3R respectively, and same charge q. Which of the spheres will have greater capacitance ?

Options

A. A
B. B
C. C
D. None of the above

Answer

To determine which of the three isolated metal spheres (A, B, and C) has the greatest capacitance, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Capacitance of a Sphere**:
The capacitance (C) of an isolated metal sphere is given by the formula:
\[ C = 4 \pi \epsilon_0 R \]
where \( R \) is the radius of the sphere and \( \epsilon_0 \) is the permittivity of free space.

2. **Identify the Radii of the Spheres**:
- Sphere A has a radius \( R \).
- Sphere B has a radius \( 2R \).
- Sphere C has a radius \( 3R \).

3. **Calculate the Capacitance for Each Sphere**:
- For Sphere A:
\[ C_A = 4 \pi \epsilon_0 R \]
- For Sphere B:
\[ C_B = 4 \pi \epsilon_0 (2R) = 8 \pi \epsilon_0 R \]
- For Sphere C:
\[ C_C = 4 \pi \epsilon_0 (3R) = 12 \pi \epsilon_0 R \]

4. **Compare the Capacitances**:
- From the calculations:
- \( C_A = 4 \pi \epsilon_0 R \)
- \( C_B = 8 \pi \epsilon_0 R \)
- \( C_C = 12 \pi \epsilon_0 R \)

Clearly, \( C_C > C_B > C_A \).

5. **Conclusion**:
The sphere with the greatest capacitance is Sphere C, which has the largest radius of \( 3R \).

### Final Answer:
Sphere C has the greatest capacitance.