Question

Consider a hollow spherical conductor with total charge \(+5 e\). The outer and inner radii are \(a\) and \(b,\) respectively. (a) Calculate the charge on the sphere's inner and outer surfaces if a charge of \(-3 e\) is placed at the center of the sphere. (b) What is the total net charge of the sphere?

Short answer

Answer: The charges on the inner and outer surfaces of the hollow spherical conductor are +3 e and +2 e, respectively. The total net charge of the sphere is +2 e.

Step-by-step solution

(a) Charge on the inner and outer surfaces

Due to the spherical symmetry of the problem, we can safely assume that the charge will distribute evenly on both the outer and the inner surfaces of the conductor. 1. First, we must note that since the sphere is a conductor and the charge within the hollow region is -3 e, the charge distribution inside the hollow region must cancel out -3 e. As a result, we can conclude that the inner surface will have a charge equal to +3 e. 2. Now, we know that the total charge of the conductor is +5 e. As we have seen, the inner surface has a charge of +3 e. The remaining charge must be distributed on the outer surface. Therefore, the charge on the outer surface is: (+5 e) - (+3 e) = +2 e. So, the charges on the inner and outer surfaces of the hollow spherical conductor are +3 e and +2 e, respectively.

(b) Total net charge of the sphere

To find the total net charge, we must consider both the charges on the inner and outer surfaces of the conductor and the charge placed at its center. To calculate the net charge, we will simply sum up these three quantities: Total net charge = Charge on inner surface + Charge on outer surface + Central charge = (+3 e) + (+2 e) + (-3 e) The net charge of the sphere is therefore equal to +2 e.