Question

4. A spherical region of space of radius 0.1 m has a volumetric charge density ρυ-kr2 where k = 4 × ttviyiteo everywhere Calculate the electric flux density (D) at r-0.02m Determine the distance r outside the sphere at which the magnitude of the electric flux density is equal to what you found in part (a). a. b.

Answer 1

r = 0.1 m [rho_v = kr^2] rho_v = 0 for r > 0.1 m volume change density, rho_v = kr^2 where, k = 4 times 10^- 5 c/m^- 5 vol^m of the elemental sphere dv = 4 pi r^2 dr dq = rho_v dv dq = kr^2 4 pi r62 dr integral^Q_0 dq = 4 pi k integral^r_0 r^4 dr Q = 4/5 pi k k (r^5) total charge of spherical Q = 4/5 pi times 4 times 10^- 5 times (0.1)^5 = 1 times 10^- 9 C (a) q_in = integral^r = 0.02_0 dq = 4 pi k integral^r_0 r^4 dr = 4 times 3.14 times 4 times 10^5 times (0.02)^5/5 = 3.21 times 10^- 13 C For gauss's law E^- . A^- = q_in/epsilon_0 where - A = 4 pir^2 at r = 0 0.02 m A = 4 times pi times (0.02)^2 D^- = epsilon_0 E^- = epsilon_0 times q_in/epsilon A^- = 3.21 times 10^- 13/4 pi times (0.02)^2 D^rightarrow = 6.38 times 10^- 11 c/m^2