Two identical non-conducting spherical shells having equal charge Q, which is uniformly distributed on it, are placed at a distance d apart. From where they are released.Find out kinetic energy of each sphere when they are at a large distance.

Answer

To solve the problem, we need to find the kinetic energy of each of the two identical non-conducting spherical shells with charge Q when they are at a large distance apart after being released from an initial separation of distance d.

### Step-by-Step Solution:

1. **Understand the System**:
- We have two identical non-conducting spherical shells, each with charge Q, placed at a distance d apart.
- The charges are uniformly distributed on each shell.

2. **Calculate Initial Potential Energy**:
- The potential energy (U) of the system when the shells are at distance d can be calculated using the formula for the potential energy of two point charges:
\[ U = \frac{kQ^2}{d} \]
- Here, \( k \) is the Coulomb's constant.

3. **Consider Self-Energy**:
- Each shell also has its own self-energy, which can be calculated using the formula:
\[ U_{self} = \frac{kQ^2}{2R} \]
- Since there are two shells, the total self-energy is:
\[ U_{self\ total} = 2 \cdot \frac{kQ^2}{2R} = \frac{kQ^2}{R} \]

4. **Total Initial Potential Energy**:
- The total initial potential energy of the system when the shells are at distance d is:
\[ U_{initial} = U + U_{self\ total} = \frac{kQ^2}{d} + \frac{kQ^2}{R} \]

5. **Final Potential Energy at Large Distance**:
- When the shells are at a large distance apart, the potential energy approaches zero because the interaction between the charges diminishes.
- The final potential energy \( U_{final} = 0 \).

6. **Apply Conservation of Energy**:
- According to the law of conservation of energy:
\[ \text{Initial Kinetic Energy} + \text{Initial Potential Energy} = \text{Final Kinetic Energy} + \text{Final Potential Energy} \]
- Initially, the kinetic energy is zero (the shells are at rest), so we have:
\[ 0 + U_{initial} = K_{final} + 0 \]
- Therefore, the final kinetic energy of the system (K) is equal to the initial potential energy:
\[ K = U_{initial} = \frac{kQ^2}{d} + \frac{kQ^2}{R} \]

7. **Distributing Kinetic Energy**:
- Since the kinetic energy is shared equally between the two shells, the kinetic energy of each shell \( K_{each} \) is:
\[ K_{each} = \frac{K}{2} = \frac{1}{2} \left( \frac{kQ^2}{d} + \frac{kQ^2}{R} \right) \]

### Final Answer:
The kinetic energy of each sphere when they are at a large distance apart is:
\[ K_{each} = \frac{1}{2} \left( \frac{kQ^2}{d} + \frac{kQ^2}{R} \right) \]