Inductance per unit length of coaxial transmission line

In summary, the author attempts to solve a homework problem by using energy, which he then states is the same as flux. He explains that current is a function of radius, and that to get the total flux, you need to cover the entire area.
  • #1
Abdulwahab Hajar
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1. Homework Statement

An air coaxial transmission line has a solid inner conductor of radius a and a very thin outer conductor of inner radius b. Determine the inductance per unit length of the line.

Homework Equations


the book states the methodology to find the inductance as follows:
1) Choose an appropriate coordinate system for the given geometry.
2) Assume a current I in the conducting wire.
3) Find B from I by Ampere's circuital law, if symmetry exists; if not Biot-Savart law must be used.
4)Find the flux linking with each turn, Φ, from B by integration:
Φ = ∫B.ds
Where S is the area over which B exists and links with the assumed current.
I couldn't really grasp quite understand what S is in this question.
5)Find the flux linkage Λ by multiplying Φ by the number of turns.
6) Find L by taking the ratio L = Λ / I

The Attempt at a Solution


I have the solution uploaded as it is in the book, I understand what happened first cylindrical coordinates were used, Current I was assumed and B was found using Ampere's law (steps 1,2, and 3).
In the 4th step which is finding the flux he integrated over the surface (1(unit length) * dr)... But then he said that represents only dΦ in other words the entire flux and commenced to integrate over dr again...
We already integrated over thus measuring the flux over the whole wire why did he integrate again??/ Thank you very much
 
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  • #2
Confession: this topic is entirely new to me, just going by the text you reproduced.

The linkage is between current at any two radii, r and s say. So it is a double integral ##\int_{s=0}^b\int_{r=0}^s##.
 
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  • #3
Abdulwahab Hajar said:
View attachment 111715 View attachment 111716 View attachment 111718 But then he said that represents only dΦ in other words the entire flux and commenced to integrate over dr again...
We already integrated over thus measuring the flux over the whole wire why did he integrate again??/
Thank you very much
dΦ is the flux over a thin rectangular strip of unit length times dr. So to get the total flux you need to cover the entire area which is a rectangle of unit length times b.

Carefrul when you compute I in the region 0<r<a; there the current is a function of r (whereas for a<r<b it's constant).

Here's a very nice diagram of the area to be integrated over:
https://www.physics.byu.edu/faculty/christensen/Physics%20220/FTI/32%20Inductance/32.11%20The%20inductance%20of%20a%20coaxial%20cable.htm

But they assume the current in the inside conductor is all on its surface, not uniformly distributed within 0<r<a as in your problem.
 
  • #4
An alternative & perhaps clearer view is go with energy:

Energy dU in a differential volume dV of a B field in vacuo is 1/2 BH dV = 1/2B2dV/μ0. Integrate B over the volume of the field, then equating U to the energy of an inductor = 1/2 LI2, getting the same answer. The total field volume is πb2 times unit length. Again, care must be taken in computing the B field in regions 0<r<a and a<r< b.
 

Related to Inductance per unit length of coaxial transmission line

1. What is inductance per unit length of a coaxial transmission line?

Inductance per unit length refers to the amount of inductance, or opposition to change in current, that is present in a certain length of a coaxial transmission line. It is commonly measured in Henrys per meter (H/m).

2. How is inductance per unit length calculated?

The inductance per unit length of a coaxial transmission line can be calculated using the formula L = μ/2π * ln(b/a), where L is the inductance per unit length, μ is the permeability of the medium between the conductors, b is the radius of the outer conductor, and a is the radius of the inner conductor.

3. What factors affect the inductance per unit length of a coaxial transmission line?

The inductance per unit length of a coaxial transmission line is affected by the spacing between the conductors, the diameter of the conductors, the material of the conductors, and the permeability of the medium between the conductors.

4. How does inductance per unit length impact signal transmission in a coaxial cable?

Inductance per unit length is an important factor to consider in signal transmission in a coaxial cable because it can cause losses and distortions in the signal. Higher inductance per unit length can result in higher losses and decrease the efficiency of the cable.

5. Can the inductance per unit length of a coaxial transmission line be adjusted?

Yes, the inductance per unit length of a coaxial transmission line can be adjusted by altering the physical characteristics of the cable, such as changing the spacing or diameter of the conductors. This can be useful in optimizing the cable for specific applications or reducing signal losses.

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