 #1
AndrewC
 6
 0
 Homework Statement

A cylindrical conducting rod of radius a = 1 cm has a nonuniform current density ๐ฑ(๐) = ๐z J0 ๐^(r/a)^2 (A/m2) and is surrounded by a cylindrical conducting surface of radius b = 10 cm carrying a current I0 in the opposite (az) direction. The region between the two conductors is filled with a material having conductivity sigma = 0 and ๐r = 100, whereas ๐r = 1 for the conductors. Assuming J0 = 1 x 10^4 A/m2 and I0 = 1 A, find:
a) The magnetic field intensity H, flux density B and magnetization M for r < a
b) The magnetic field intensity H, flux density B and magnetization M for a < r < b
c) The magnetic field intensity H, flux density B and magnetization M for r > b
 Relevant Equations

Amperes circuital law:
โฎ๐โd๐ฅ= ๐0 ๐ผ๐๐๐
โฎ๐โd๐ฅ= ๐ผ๐๐๐
Magnetization:
๐= ๐๐0โ๐
๐=๐๐ ๐
Inner conductor radius = 1cm
outer conductor radius = 10cm
region between conductors has conductivity = 0 & ๐r = 100
๐r = 1 for inner and outer conductor
Io = 1A(az)
๐ฑ(๐) = (10^4)(๐^(r/a)^2)(az)
Problem has cylindrical symmetry, use cylindrical coordinate system.
Find the total current enclosed by inner conductor for r<a:
Ienc = (0,r)โซ (10^4)(๐^(r/a)^2)(2ฯr)dr
= 2ฯ*10^4(โซ(r๐^(r/a)^2)dr
let t = (r/a)^2, dt = (2rdr)/a^2
Ienc = a^2(ฯ*10^4)โซ(e^t)dt from 0 to โt(a^2)
Ienc = a^2(ฯ*10^4)[e^t] from 0 to โt(a^2)
At this point I started questioning whether I was doing this right. Would appreciate any pointers on proper setup of amperes law.
outer conductor radius = 10cm
region between conductors has conductivity = 0 & ๐r = 100
๐r = 1 for inner and outer conductor
Io = 1A(az)
๐ฑ(๐) = (10^4)(๐^(r/a)^2)(az)
Problem has cylindrical symmetry, use cylindrical coordinate system.
Find the total current enclosed by inner conductor for r<a:
Ienc = (0,r)โซ (10^4)(๐^(r/a)^2)(2ฯr)dr
= 2ฯ*10^4(โซ(r๐^(r/a)^2)dr
let t = (r/a)^2, dt = (2rdr)/a^2
Ienc = a^2(ฯ*10^4)โซ(e^t)dt from 0 to โt(a^2)
Ienc = a^2(ฯ*10^4)[e^t] from 0 to โt(a^2)
At this point I started questioning whether I was doing this right. Would appreciate any pointers on proper setup of amperes law.