Problem Statement

Two identical non-conducting spherical shells, each having a charge Q uniformly distributed on it, are placed at a distance D apart and then released. We need to find the kinetic energy of each sphere when they are at a large distance.

Solution

Understanding the Problem

To solve this problem, we need to consider the electric potential energy and the principle of conservation of energy. Initially, the two spheres are at rest and have only electric potential energy. As they move apart, this potential energy is converted into kinetic energy.

Electric Potential Energy

Electric potential energy is given by the equation:

U = k * (q1 * q2) / r

Where:
- U is the electric potential energy
- k is the Coulomb constant
- q1 and q2 are the charges on the two spheres
- r is the distance between the centers of the spheres

Conservation of Energy

According to the principle of conservation of energy, the sum of the initial potential energy and the initial kinetic energy is equal to the sum of the final potential energy and the final kinetic energy.

Initial energy = Final energy

Initially, the spheres are at rest, so their initial kinetic energy is zero.

Initial energy = Electric potential energy

Finally, when the spheres are at a large distance, the electric potential energy approaches zero. Therefore, the final energy is equal to the final kinetic energy.

Final energy = Final kinetic energy

Calculating the Final Kinetic Energy

Using the conservation of energy principle, we can write:

Initial energy = Final energy

Electric potential energy = Final kinetic energy

Using the equation for electric potential energy, we have:

k * (q1 * q2) / r = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2

Since the spheres are identical, their masses (m1 and m2) are equal. Also, their velocities (v1 and v2) are equal as they are connected by a rigid rod.

Simplifying the equation, we get:

k * (q1 * q2) / r = m * v^2

Where:
- m is the mass of each sphere
- v is the velocity of each sphere

Now, we can solve for the kinetic energy:

Kinetic energy = (1/2) * m * v^2

Therefore, the kinetic energy of each sphere when they are at a large distance is given by:

Kinetic energy = (1/2) * m * v^2

Conclusion

In conclusion, the kinetic energy of each sphere when they are at a large distance is given by (1/2) * m * v^2, where m is the mass of each sphere and v is the velocity of each sphere. This is derived from the conservation of energy principle, considering the initial electric potential energy and the final kinetic energy.