Question

4 A spherically symmetric charge distribution has the following radial dependence for the volume charge density ρ: 0 if r R where γ is a constant a) What units must the constant γ have? b) Find the total charge contained in the sphere of radius R centered at the origin c) Use the integral form of Gauss's law to determine the electric field in the region r R. (Hint: if the charge distribution is spherically symmetric, what can you say about the electric field?) d) Repeat part c) using the differential form of Gauss's law (you may again simplify the calculation with symmetry arguments e) Using any method of your choice, determine the electric field in the region r> R. f) Suppose we wish to enclose this charge distribution within a hollow, conducting spherical shell centered on the origin with inner radius a and outer radius b (R < < b) such that the electric field for the region r > b is zero. In this case. what is the net charge carried by the spherical shell How much charge is located on the inner radius a and the outer radius rb? What is the electric field in the regions r < R, R<r<a, and a Kr b? For part f), I am not looking for a lot of additional work. Just use your answers to the previous parts, Gauss's law, and what you knoui a out electric ields tn side conductors or static systems饥order to erplaan your answers.

Answer 1

Dim of rho = c/m^3 gamma/r^2 = c/m^3 gamma = c/m (units of gamma) q_T = integral^infinity_0 f(gamma) dr 4 pi r^2 = 4 pi integral^R_0 gamma/r^2 e^-r/R r^2 dr = 4 pi gamma integral^R_0 e^-r/R dr q_T = 4 pi R gamma[1 - e^-1] contourintegral E^vector middot ds^vector = q_ens/E_0 E times 4 pi gamma^2 = 4 pi gamma/E_0 integral^gamma_0 e^-r/R 1/r^2 r^2 dr E(r) = R gamma/r^2 E_0[1 - e^-r/R] \r\n nabla middot E = rho/E_0 1/r^2 partial differential/partial differential r(r^2 E_r) = gamma/r^2 E_0 e^-r/R only E_r exists) r^2 E_r = gamma/E_0 integral^r_0 e^-r/R E_(r) = R gamma/E_0 r^2[1 - e^-r/R] E times 4 pi r^2 = q_T/E_0 E times 4 pi r^2 = 4 pi R gamma/E_0[1 - e^-1] E = R gamma/r^2 E_0[1 - e^-1]