Question

energy of configuration

Problem 4 (5 pts): A sphere of radius Ro carries a charge density ρ (R)-R where k is some constant(not equal to 41e). Find the energy of the configuration using: V dv

Answer 1

rho (r) = kr k notequalto 1/4 pi epsilon_0 We know that Electro field inside and outside the sphere, using Gauss�s law E_0 contourintegral E. da = E_0 4 pi r^2 E = Q_enclosed Q_enclosed = integral p dr = integral kr r^2 sin theta dr^^d theta d Phi = 4 pi k integral^r^^_0 r^- 3 dr^^= {pi k r^4 (r < R) pi k R^4 (r > R) So E = k pi r^4 r^^4 pi r^2 E_0 (r < R) E = k R^4/ E_0 r^2 r^^r > R W = 1/2 integral E_0 E^2 dr = E_0/2 integral^R_0 (k r^2/4 E_0)^2 4 pi r^2 dr + E_0/2 integral^infinity_R (k R^4/4 E_0 r^2)^2 4 pi r^2 = 4pi E_0/2 (k/4 E_0)^2 integral^R_0 r^6 dr + E_0/2 4 pi [R^8] (k/4 pi E_0)^2 integral^infinity_R r^2/r^2 dr = 4 pi E_0/2 (k/4 E_0)^2 [integral^R_0 r^6 dr + R^8 [integral^infinity_R 1/r^2 dr]] = pi k^2/8 E_0 [R^7/7 + R^R [1/R] W = pi k^2 R 8/7 E_0