Question

A conducting spherical shell with inner radius ‘a’ and outer radius ‘b’ has a positive charge Q located at its center. The total charge on the shell is -4Q, and it is insulated from its surroundings.

(a) Derive expressions for the electric field magnitude in terms of the distance ‘r’ from the center for the regions: r , a; a < r < b; r >b.

Answer 1

The electric field expressions for regions within and around a conducting spherical shell with inner radius 'a', outer radius 'b', and centered charge Q are: E = 0 for r < a; E = kQ/r^2 for a < r < b; and E = k(-3Q)/r^2 for r > b, derived from Gauss's Law.

Step-by-step explanation:

To derive expressions for the electric field magnitude in terms of the distance 'r' from the center of a conducting spherical shell with inner radius 'a' and outer radius 'b', one can make use of Gauss's Law. Considering the charge Q at the center and total charge -4Q on the shell:

For r < a (inside the inner radius), the electric field E = 0 since the electric field inside a conductor in electrostatic equilibrium is zero.

For a < r < b (between the inner and outer radius), the enclosed charge by a Gaussian surface is Q, so using Gauss's Law, E = kQ/r^2, where k is the electrostatic constant.

For r > b (outside the outer radius), the total enclosed charge is Q - 4Q = -3Q, so E = k(-3Q)/r^2.

This covers the regions specified with their respective electric field expressions based on the radial distance 'r' from the center.