Question

A thick spherical shell (inner radius a, outer radius b) is made of dielectric material with a "frozen-in" polarization P(r) 0 r<a P(r) ksin(0)/r r a<r<b where k is a constant, r is the distance from the center, and r is the radial unit vector. There is no free charge in the problem 1. Find expressions for all the bound (volume and surface) charge. Interpret with a diagram. 2. Determine the total bound charge. Be aware if the bound charge densities depend on r, ф or you must integrate with respect to those variables to find the total charge enclosed 3. Use (where possible) the Displacement vector approach to find the electric field in the regions r<a, a<r<b, r>b. If not possible explain why

Answer 1

Given that The bound volume charge is, P_b = -nabla middot P^- = -1/r^2 middot Partialdifferential/Partialdifferential r (r^2 P_r) = -1/r^2 Partialdifferential/Partialdifferential r (kr) = -k/r^3 on the outer surface, r = b and n^^= r^^so the bound surface charge there is sigma_b = P vector middot n^^= (k/b r^^) middot r^^= k/b on the inner surface, r = a and n^^= -r^^, so, the bound surface charge there is, sigma_k = P^^middot n^^= -k/a The problem is spherically symmetric, so, E vector = E (r) r^^and a sphere of radius r is a good choice for a Gaussian surface Regardless of the radius, the electric flux is contourintegral E da = E 4 pi r^2 the charge enclosed is Q_enc = 0 For r < a and r > b (net bound charge = 0) E = 0 for r < a & r > 0