Question

Problem 4, 30 marks The infinitely long conducting cylinder of radius R carries the volume current density directed along its axis whose absolute value is a cubic function of the distance from the center of the cylinder r, j(r)-br3, where b is a known constant. a. Find the magnitude and direction of the magnetic field B forr>R. b. Find the magnitude and direction of the magnetic field B for r<R. c. Imagine that the conductor has magnetic permeability H (5 marks) (5 marks) (i) Find magnitudes and directions of the vectors H, B, M for r<R 5 marks) (ii) Find the surface current density of the bound current Kb at the surface of the cylinder (5 marks) (iii) Find the volume current density of the bound current Jb inside the cylinder (10 marks)

Answer 1

(a) magnetic field at r > R contourintegral B middot dl = mu_0 I_end E_enclosed = I (Because current flow in surface) B middot 2 pi r = mu_0 I (b) magnetic field r < R we know that I = integral j ds L I = integral^R_0 integral^2pi_0 br^3 dr r d phi = 2 pi b r^5/5 {where b is known constant] From (1) J (r) = br^3 = 5I/2 pi R^5 r^3 - (2) I_(enclosed) = integral^r_0 integral^2 pi_0 5I/2 pi R^5 r^3 middot r dr d phi phi implies 5I/2 pi R^5 times 2 pi r^5/5 I_(enclosed) = Ir^5/R^5 B_r < R = mu_0 Ir^5/R^5 phi (c) Suppose mu is magnetic permeability contourintegral H middot dl = I_free [But these is no free current] so H = 0 & B = mu [H+ m] = mu [0 + m] B = mu m r < R mu = mu r mu_0 mu_r mu_0 m = mu_0 I r^5/R^5 m = I r^5/R^5 mu_r