Question

A small conducting sphere A of radius \( r \) charged to a potential \( V \), is enclosed by a spherical conducting shell B of radius \( R \). If A and B are connected by a thin wire, calculate the final potential on sphere A and shell B.

Hint:

When two conductors are connected by a wire, they must be at the same potential, which is achieved by redistributing the charge between them.

Answer 1

Final Potential on Sphere A and Shell B

Given:

• Radius of sphere A: \( r \)
• Radius of shell B: \( R \)
• Initial potential on sphere A: \( V \)
• Spheres A and B are connected by a thin wire.

Solution:

1. Initial Setup:

Initially, sphere A has a charge \( Q_A \) corresponding to the potential \( V \), and shell B is uncharged. When the two conductors are connected by a wire, charges will flow between them until they reach the same potential, because a conducting wire ensures that the potential difference between them becomes zero.

2. Charges and Potentials:

The potential of a sphere due to its charge is given by the formula: \[ V = \frac{kQ}{r} \] where: - \( k \) is Coulomb's constant \( \left( k = \frac{1}{4 \pi \epsilon_0} \right) \), - \( Q \) is the charge on the sphere, - \( r \) is the radius of the sphere.

3. Charge on Sphere A:

The initial charge on sphere A, when its potential is \( V \), is: \[ Q_A = \frac{V r}{k} \]

4. Final Situation:

When spheres A and B are connected, charge will flow until both spheres are at the same potential. Let \( V_f \) represent the final common potential of both sphere A and shell B.

5. Total Charge:

Since charge is conserved, the total charge on both sphere A and shell B must remain the same. The total charge initially on the system is just the charge on sphere A (since shell B starts uncharged): \[ Q_{\text{total}} = Q_A = \frac{V r}{k} \]

6. Potential on Sphere A and Shell B:

The potential of sphere A after charge redistribution is: \[ V_A = \frac{k Q_A}{r} \] And the potential of shell B after redistribution is: \[ V_B = \frac{k Q_B}{R} \] Since \( Q_A = Q_B \), we can say that the final potential \( V_f \) on both spheres is: \[ V_f = \frac{k Q_A}{R} \]

7. Final Potential Calculation:

Using the fact that the total charge is conserved, and that the potential on both spheres is the same, we get the final potential: \[ V_f = \frac{k Q_A}{r + R} \]

Conclusion:

The final potential on sphere A and shell B is \( V_f \), and it is the same for both: \[ V_f = \boxed{\frac{V r}{R}} \]