Question

An infinitely long circular cylinder carries a permanent magnetization M = ks^2 zˆ.

a) Calculate the bound current densities Jb and Kb.

b) Calculate the total current due to Jb circulating around the axis of the cylinder within a section of length ∆z = L, and indicate its direction. (Show your integral explicitly.)

c) Calculate the total current due to Kb circulating around the axis of the cylinder within a section of length ∆z = L, and indicate its direction. How does your answer compare with part (b)?

d) Calculate the B-field B inside the cylinder. Explain carefully how you determined both its direction and magnitude.

e) Calculate the H-field H inside the cylinder, first by using the fundamental relation B =µ0(H+M), and second by using Ampere’s law. In the latter case, draw a diagram showing the integration loop, and carefully explain your integration and any symmetries you assumed.

Answer 1

ka) Calculate the volume bound current as follows: ja = VⓇM =-0 (2 ks) Hence, the volume bound current is –2ksø. Also, the surface bound current is calculated as follows: K= M N = ks 2 x Š = ks Hence, the surface bound current is ks?ộ.

(b) Now, calculate the total current due to volume bound current density as follows: 16 = Sjo. da =-5J(-2ksộ. bald:) -70 =-kal Hence, the total bound current is – ka’L.

Now, calculate the total current due to bound surface current density as follows: Is'= Kg dữ a 21 = ka' || ado 0 0 = 2nka Hence, the total bound current due to bound surface current density is 2nka.

(d) Now, calculate the magnetic field inside the cylinder as follows: $ # dī = Igie $ # •dī = 0 Thus, Ã = 0 Finally, B = H. (H + M) = 16 (0+M) = H0 (ks :) = 4 ks Hence, the magnetic field is unks-s.