- #1

AndrewC

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- Homework Statement
- A cylindrical conducting rod of radius a = 1 cm has a non-uniform 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.