Question

2. A solid cylinder of radius R1 and permeability u1 has a uniform surface current density K = K2 at its surface, where 2 is the axis of the cylinder. The cylinder is covered by a coaxial cylindrical shell of inner radius R1, outer radius R2 made of a material of permeability H2. The cylindrical shell has no free current. a) Calculate H, B and M in all three regions 0 < r < R1, R1 < r < R2, and R2 < r < 0 Note that since the system is translationally invariant in the z-direction and symmetric about the z-axis the fields do not depend on the z and φ coordinates in the cylindrical b) c) coordinate system. Calculate the bound volume current density Jb in all material regions. Calculate the bound surface current density K, at r-R1 and at r-R2 d) Calculate the magnetostatic energy per unit length in the z-direciton using Umag- e) Calculate the magnetostatic energy per unit length in the z-direction using Umag H-Bdr. Is there any difference from the result of (d)?

Answer 1

Now, H^vector = B^vector/mu so H^vector = {0, 0 lessthanorequalto r lessthanorequalto R_1 kR_1/r phi^^, R_1 lessthanorequalto r lessthanorequalto R_2 kR_1/r phi^^, r > R_2 then, M = chi_m H (chi_m = mu_2/mu_0 - 1) M^vector = {0, 0 lessthanorequalto r lessthanorequalto R_1 chi_m kR_1/r phi^^, R_1 lessthanorequalto r lessthanorequalto R_2 chi_m kR_1/r phi^^, R_2 lessthanorequalto R b) J_b^vector = nabla^vector times M^vector = 1/r Vertical-bar r^^partial differential_r 0 r phi^^partial differential phi chi_m kR_1/r z^^partial differential z 0 Vertical-bar = 1/r z^^[-X_m kR_1/r^2] J_b^vector = -chi_m kR_1/r^3 z^^ \r\n Now, H^vector = B^vector/mu so H^vector = {0, 0 lessthanorequalto r lessthanorequalto R_1 kR_1/r phi^^, R_1 lessthanorequalto r lessthanorequalto R_2 kR_1/r phi^^r > R_2 then, M = chi_m H (X_m = mu_2/mu_0 - 1) M^vector = {0, 0 lessthanorequalto r lessthanorequalto R_1 chi_m kR_1/r phi^^, R_1 lessthanorequalto R lessthanorequalto R_2 chi_m kR_1/r phi^^, R_1 lessthanorequalto r b) J_b^vector = nabla^vector times M^vector = 1/r Vertical-bar r^^partial differential_r