# Flux linkage in inductance calculation for single wire.

• goodphy
In summary, the conversation discusses the concept of flux linkage in relation to Ampere's law and the calculation of inductance for a single round wire. It is clarified that flux linkage only considers the flux within a circuit and not the external field. The concept of external flux linkage is also mentioned, but it is not typically considered in the calculation of inductance. The conversation also touches on the importance of considering the return current path in the calculation of inductance for realistic models. Finally, the concept of total flux being zero due to the absence of magnetic monopoles is explained.
goodphy
Hello.

Maybe this is the one of the most typical example in Electromagnetism textbook.

There is a single round wire carrying current I (in D.C) with radius of R and length is infinity.

From Ampere's law, the B field inside the wire is

B = μIρ/2πR2 aθ

where ρ is radial distance from symmetric axis to point of field and aθ is unit vector in azimuthal coordinate in cylindrical coordinate.

The magnetic flux through the region of infinitesimal area of width dρ and unit length is simply

1 = Bdρdz = μIρ/2πR2dρdz.

It is very straightforward until this step.

My question arise from here; According to textbook, the flux linkage is not equal to flux but flux multiplied Ienc/I. It seems flux linkage counts only contribution of current which actually induces B in flux above. It seems reasonable since the flux linkage should not count the current which doesn't contribute.

Physically I feel it make sense, but I need a more rigorous definition of flux linkage to redrive this mathematical expression.

Could you please help me how to understand the flux linkage in clear view?

And in addition inductance L for single round wire is only calculated with the flux inside the conductor. But still there is a field outside the wire! Why is external field not counted?? Maybe this field is associated to external inductance Lext and total inductance L = Lint + ext where Lint is linked to flux linkage inside the wire as described above?

flux linkage is defined as the surface integral "flux" of the B field within a circuit. From this, is it clear why would external field lines be relevant?

Stephen Hodgson said:
flux linkage is defined as the surface integral "flux" of the B field within a circuit. From this, is it clear why would external field lines be relevant?
Thanks for giving some comment. According to your reply, the flux linkage is the concept only associated to 'within' the circuit, rather than considering the external to it. But it has a field outside which also store the energy. Thus I guess there should be a concept of like 'external flux linkage' taking this uncounted energy storing field into account.

At least flux linkage problem, I've found the general definition of the flux linkage as ΛI = ∫IΛdΨ where Λ is flux linkage, dΨ is the differential flux due to IΛ, actual fraction of current which dΨ is due to. This is also applicable to coil in which ∫IΛdΨ becomes IΛΨ since magnetic field strength B is considered as uniform inside the coil. Here IΛ should be IN where N is number of turns in coil because Ψ is associated (linked) to all current IN.

Please see http://www.ece.mcmaster.ca/faculty/nikolova/EM_2FH3_downloads/lectures/L19_Induct_post.pdf

It comes to that I only need to solve why external inductance (the inductance due to field outside the conductor) is not typically counted in straight wire.

I had a wrong idea of wire selection in thought. What I'd been stuck is the fact that external inductance is infinity for single wire. It means single wire only passes DC, not true in real world. In fact, I've found that two parallel wires are more realistic model as there should be always other path, return current path. Two parallel wires are well known and I finally got rid of single wire in my mind.

Yeah, the problem is, you don't know what the circuit is. If we had 2 parallel wires, we could work out the inductance (per unit length) from the flux between the wires. A single wire on its own bears no associated inductance. Thus the inductance associated with a wire of finite radius alone would exclude this.

goodphy said:
Thanks for giving some comment. According to your reply, the flux linkage is the concept only associated to 'within' the circuit, rather than considering the external to it. But it has a field outside which also store the energy. Thus I guess there should be a concept of like 'external flux linkage' taking this uncounted energy storing field into account.

The total flux is zero. This is because there are no magnetic monopoles. So measured across all space, for every flux line adding, there will be one somewhere subtracting.

Flux linkage is thus the total amount of flux through a region. Usually the region is well defined by some wires (like a coil), but it doesn't need to be.

In the case of your long, straight wire, maybe a box with an edge along the surface of the wire a unit length long, and two edges stretching to infinity?

## 1. What is flux linkage?

Flux linkage refers to the amount of magnetic flux that passes through a given area. It is a measure of the total magnetic field that is linked to a particular circuit or component.

## 2. Why is flux linkage important in inductance calculation for single wire?

In inductance calculation for single wire, flux linkage is important because it is a key factor in determining the amount of inductance a wire or circuit has. The more flux linkage, the higher the inductance.

## 3. How is flux linkage calculated?

Flux linkage is calculated by multiplying the magnetic flux by the number of turns in a coil or circuit. This can be represented by the equation Φ = B * N, where Φ is the flux linkage, B is the magnetic flux, and N is the number of turns.

## 4. What factors affect flux linkage in inductance calculation for single wire?

The factors that affect flux linkage in inductance calculation for single wire include the number of turns in the wire or circuit, the strength of the magnetic field, and the area through which the flux passes. Additionally, the material of the wire and the distance between the wire and the magnetic field source can also impact flux linkage.

## 5. How does flux linkage impact the inductance of a single wire or circuit?

Flux linkage is directly proportional to the inductance of a single wire or circuit. This means that as the flux linkage increases, so does the inductance. This is because a higher flux linkage indicates a stronger magnetic field, which in turn can store more energy in the form of inductance.

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