Magnetic field intensity, flux density and magnetization of coax cable

In summary, we have a problem with cylindrical symmetry and use cylindrical coordinate system. We are asked to find the total current enclosed by the inner conductor for r<a. Using Amperes law and area integration, we calculate the total current to be Ienc = ฯ€a^2(โˆซ0^1 e^-r^2 d(r^2)).
  • #1
AndrewC
6
0
Homework Statement
A cylindrical conducting rod of radius a = 1 cm has a non-uniform current density ๐‘ฑ(๐‘Ÿ) = ๐’‚z 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 ๐œ‡r = 100, whereas ๐œ‡r = 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:
โˆฎ๐โˆ™d๐ฅ= ๐œ‡0 ๐ผ๐‘’๐‘›๐‘
โˆฎ๐‡โˆ™d๐ฅ= ๐ผ๐‘’๐‘›๐‘
Magnetization:
๐‡= ๐๐œ‡0โˆ’๐Œ
๐Œ=๐œ’๐‘š ๐‡
Inner conductor radius = 1cm
outer conductor radius = 10cm
region between conductors has conductivity = 0 & ๐œ‡r = 100
๐œ‡r = 1 for inner and outer conductor
Io = 1A(-az)
๐‘ฑ(๐‘Ÿ) = (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)โˆซ (10^4)(๐‘’^-(r/a)^2)(2ฯ€r)dr

= 2ฯ€*10^4(โˆซ(r๐‘’^-(r/a)^2)dr

let t = (r/a)^2, dt = (2rdr)/a^2

Ienc = a^2(ฯ€*10^4)โˆซ(e^-t)dt from 0 to โˆšt(a^2)

Ienc = a^2(ฯ€*10^4)[-e^-t] from 0 to โˆšt(a^2)

At this point I started questioning whether I was doing this right. Would appreciate any pointers on proper setup of amperes law.
 
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  • #2
Hi. Area integration would be
[tex]\int_0^a e^{-(r/a)^2} dA

= 2\pi \int_0^a e^{-(r/a)^2} r dr

= 2\pi a^2 \int_0^1 e^{-r^2} r dr

= \pi a^2 \int_0^1 e^{-r^2} d(r^2)=...[/tex]
 

What is magnetic field intensity?

Magnetic field intensity is a measure of the strength of a magnetic field at a specific point. It is typically measured in units of amperes per meter (A/m) or tesla (T).

What is flux density?

Flux density, also known as magnetic flux density or magnetic induction, is a measure of the amount of magnetic field passing through a given area. It is typically measured in units of tesla (T) or gauss (G).

What is magnetization?

Magnetization is the process of aligning magnetic dipoles within a material in order to create a net magnetic field. It is typically measured in units of amperes per meter (A/m) or tesla (T).

How do these properties relate to coaxial cables?

Coaxial cables have a central conductor surrounded by an insulating layer and an outer conductor. The magnetic field generated by the flow of current through the central conductor induces a magnetic field in the outer conductor. The magnetic field intensity, flux density, and magnetization of the coaxial cable are important factors in determining its performance and efficiency.

What factors affect the magnetic properties of coaxial cables?

The magnetic properties of coaxial cables can be affected by the materials used, the geometry of the cable, and the frequency of the signals being transmitted. The type and thickness of the insulating and outer conductor materials can also impact the magnetic properties of the cable.

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