Question

3 SOLID SPHERE Consider a solid sphere of radius R with charge per unit volume that depends only on the distance from the origin, r, 3.1 15 POINTS Compute the electric field everywhere inside the sphere. Be sure to state the magnitude and direction of E as a function of position within the sphere. 3.2 10 POINTS Compute the electric field everywhere outside the sphere.

Answer 1

consider this r < R₁ <R, r is a variable

consider a shell of radius r and thickness dr

Consider this r < R_1 < R, r is a variable consider a shell of radius r and thickness dr Volume of the shell is 4 pi r^2 times dr charge density is rho(r) = rho_0 R/r charge in this shell is 4 pi r^2 dr times rho_0 R/r = 4 pi rho_0 R r dr integrate from 0 to R_1 integral^R_1_0 4 pi rho_0 R r dr = 4 pi rho_0 R integral^R_1_0 r dr = 4 pi rho_0 R[r^2/2]^R_1_0 = 4 pi rho_0 R[R_1^2/2 - 0] = 4 pi rho_0 R R_1^2/2 = 2 pi rho_0 R R_1^2 if R_1 is R ie, total charge in the given sphere is = 2 pi rho_0 R^3 (R_1 = R) \r\n for any point inside at a distance R_1 the electric field is KQ/R_1^2 = K/R_1^2 times 2 pi rho_0 R R_1^2 = 2 pi k rho_0 R for any point outside at a distance r from centre =