# Flux linkage in inductance calculation for single wire.

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.

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?

Matternot
Gold Member
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?

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.

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.

Matternot
Gold Member
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.

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?