A hollow sphere of radius 2R is charged to V volts and another smaller sphere of radius R is charged to V/2 volts. Now the smaller sphere is placed inside the bigger sphere without changing the net charge on each sphere. The potential difference between the two spheres would be

Options

A. `(3V)/(2)`
B. `(V)/(4)`
C. `(V)/(2)`
D. V

Answer

To solve the problem, we need to find the potential difference between two spheres: a larger hollow sphere of radius \(2R\) charged to \(V\) volts and a smaller sphere of radius \(R\) charged to \(V/2\) volts. The smaller sphere is placed inside the larger sphere without changing the net charge on either sphere.

### Step-by-Step Solution:

1. **Identify the Charges on the Spheres:**
- Let the charge on the larger sphere (radius \(2R\)) be \(Q_1\).
- The potential \(V\) of the larger sphere is given by:
\[ V = \frac{kQ_1}{2R} \quad \text{(where \(k = \frac{1}{4\pi\epsilon_0}\))} \]
- Thus, the charge \(Q_1\) can be expressed as:
\[ Q_1 = \frac{2VR}{k} \]

- For the smaller sphere (radius \(R\)), let the charge be \(Q_2\).
- The potential \(V/2\) of the smaller sphere is given by:
\[ \frac{V}{2} = \frac{kQ_2}{R} \]
- Thus, the charge \(Q_2\) can be expressed as:
\[ Q_2 = \frac{VR}{2k} \]

2. **Understanding the Configuration:**
- When the smaller sphere is placed inside the larger sphere, it creates a spherical capacitor configuration. The potential difference between the two spheres needs to be calculated.

3. **Calculate the Potential of Each Sphere:**
- The potential \(V_s1\) at the surface of the smaller sphere (radius \(R\)) is:
\[ V_s1 = \frac{kQ_2}{R} \]
Substituting \(Q_2\):
\[ V_s1 = \frac{k \cdot \frac{VR}{2k}}{R} = \frac{V}{2} \]

- The potential \(V_s2\) at the surface of the larger sphere (radius \(2R\)) is:
\[ V_s2 = \frac{kQ_1}{2R} \]
Substituting \(Q_1\):
\[ V_s2 = \frac{k \cdot \frac{2VR}{k}}{2R} = V \]

4. **Calculate the Potential Difference:**
- The potential difference \(V_{diff}\) between the two spheres is:
\[ V_{diff} = V_s2 - V_s1 = V - \frac{V}{2} = \frac{V}{2} \]

5. **Final Result:**
- Therefore, the potential difference between the two spheres is:
\[ V_{diff} = \frac{V}{2} \]

### Conclusion:
The potential difference between the two spheres is \( \frac{V}{2} \).