Question

A long, straight wire is aligned with the z-axis. It has constant linear charge density λ and is surrounded by a coaxial cylindrical shell of radius R and surface charge density σ = −λ/(2πR). The region 0 < s < R/2 is filled with a linear dielectric of electric susceptibility χe, and there is no dielectric anywhere else. Label regions of space as follows (s is the distance from the z axis): region A (0 < s < R/2), region B (R/2 < s < R), and region C (s > R).

(a) Find E, P, and D in all three regions.

(b) For each of the above three fields, explain whether there a discontinuity when moving radially

outward from s = (R/2)− to s = (R/2)+. What is the origin of the discontinuity if one exists?

(c) For each of the above three fields, explain whether there a discontinuity when moving radially

outward from s = R− to s = R+. What is the origin of the discontinuity if one exists?

Answer 1

Region A [0 < 5 < R/2] or dielectric region. By using Gauss's law. integral D.dA = q D.2pi SL = lambda L Therefore L is the length of wire & cable. D = lambda/2 pi s s^D = epsilon, epsilon_0 E Therefore E = lambda/epsilon_0 epsilon_r 2pi s s^P = epsilon_0 phi_e E P = epsilon_0 phi e lambda/epsilon_0 epsilon_r 2pi s s^P = phi e lambda/epsilon_r. 2pi s s^Region B [R/2 < s < R] gt is a air space so epsilon_r = 1 By using Gauss's law integral D. dA = q D.2 pi SL = lambda L D = lambda/2pi s s^Therefore E = lambda/2pi s epsilon_0 s^Polarization (P) P = epsilon_0 phi_e E gm air space phi_e = 0 hence, P = 0 Region (c) [S greaterthan R] By using Gauss's law. given sigma = -lambda/2pi R integral D.dA = q D.2pi SL = lambda. 2pi SL D = lambda/2pi R s^Therefore E = -lambda/2 pi epsilon R s^