A static electric charge is distributed in a spherical shell of inner radius R_1 and outer radius R_2. The electric charge density in the shell is given by rho = a + br, where r is the distance from

a. The electric field everywhere can be found using Gauss's law. Consider a Gaussian surface of radius r, where r < R1. Since the charge density is zero within the Gaussian surface, the electric flux through the surface must be zero. Therefore, the electric field inside the Gaussian surface must be zero. For {eq}R_1 < r < R_2{/eq}, we can use Gauss's law to find the electric field as follows. Let Q be the total charge within the spherical shell, and let E be the electric field at a point r from the center of the shell. Consider a Gaussian surface of radius r, enclosing a charge q = (a + br)4?r^2. Then, applying Gauss's law, we have: E(4?r^2) = Qenc / ?0 where Qenc is the charge enclosed by the Gaussian surface. For {eq}R_1 < r < R_2{/eq}, Qenc = q, so we have: E = q / (4??0r^2) Substituting q = (a + br)4?r^2, we have: {eq}E = \frac{a + br}{epsilon_0 r^2} {/eq} For {eq}r > R_2{/eq}, all the charge is within a sphere of radius R2, and the electric field can be found using the same method as above: E = Q / (4??0r^2) where Q is the total charge within the sphere of radius R2. Therefore, E = (a + bR2) / (?0r^2) Explanation: The electric field is zero in the region {eq}r < R_1{/eq} because there is no charge in this region to create an electric field. Gauss's law tells us that the electric flux through a closed surface is proportional to the charge enclosed by the surface. If there is no charge within the surface, there can be no flux and hence no electric field. b. The electric potential for {eq}r < R_1{/eq} can be calculated using the formula: V(r) - V(?) = - ? E · dr where V(?) is taken as zero. For {eq}r < R_1{/eq}, the electric field is zero, so the integral gives: V(r) = V(?) = 0 Therefore, the electric potential inside the sphere of radius {eq}R_1{/eq} is zero. The energy density for {eq}r < R_1{/eq} can be calculated using the formula: u = ?0 E^2 / 2 For {eq}r < R_1{/eq}, the electric field is zero, so the energy density is zero. Explanation: Since the electric field is zero for {eq}r < R_1{/eq}, there is no potential difference between any two points in this region. This means that the electric potential is constant and taken as zero. Since the energy density is proportional to the square of the electric field, which is zero for {eq}r < R_1{/eq}, the energy density is also zero in this region.