# Understanding Flux Linkage: Exploring Examples & Confusions

• Selectron
In summary, the concept of flux linkage involves calculating the magnetic flux density integrated over a specific area. Examples of this include the flux linkage of a coil moving through a magnetic field and the self-inductance of a coaxial cable and wire. In both cases, the area being integrated over is a rectangle, which may seem confusing at first. The justification for this is based on the specific conditions and variables involved, such as the distance from the center of the wire and the length of the cable. Ultimately, the determination of the area to integrate over when calculating flux linkage is dependent on the specific situation and variables involved.
Selectron
(N.B. All the following assumes that the magnetic fields are in vacuo).

I'm having trouble deciding what flux linkage is (as in Faraday's law etc.). What is the proper definition?

My textbook gives the example of a coil moving through a magnetic field, tracing out a cylinder - the flux linked is the magnetic flux density integrated over the ends of the cylinder. This I can understand easily enough, but two examples of flux linkage in particular confuse me:

1. Self-inductance of a coaxial cable - the text-book derivation of this calculates the flux linkage in the cable as follows:

Between the two conductors, the flux density (B) = mu0*I/(2*pi*r) - this bit I get. However, the total flux linkage is then given by:

phi = Int[B dA] = Int ab[B l*dr]

where a and b are the radii of the inner and outer conductors respectively and l is the cable length. So it seems that the area being integrated over is a rectangle that sits between the two conductors along the whole of the length of the cable. What is the justification for having this as the area?

2. Self-inductance of a wire - again from the textbook:

Inside the wire B = mu0*I*r/(2*pi*R2), where r is the distance from the centre of the wire, R is the radius of the wire and I is a constant DC current - again this bit I am fine with.

However, in calculating the total flux linked by a current filament at radius r, the text-book integrates over all the flux produced at a radius greater than r

i.e. flux linked by filament = Int rR[B l*dx]

where again l is the length of the wire. What is the justification for taking the area as being a rectangle of length l positioned between x=r and x=R (where x is the distance from the centre of the wire)? Surely going by Ampere's law it should be the current at a radius below r that is important if anything.

I guess what I am really asking is how do you determine what area to integrate over when calculating the flux linkage?

## Related to Understanding Flux Linkage: Exploring Examples & Confusions

Flux linkage is the measure of the amount of magnetic flux that passes through a circuit or device. It is represented by the symbol Φ and is measured in weber (Wb).

## How is flux linkage calculated?

Flux linkage is calculated by multiplying the number of turns in a coil by the magnetic flux passing through it. Mathematically, it can be represented as Φ = NΦ, where N is the number of turns and Φ is the magnetic flux.

## What are some examples of flux linkage in everyday life?

Flux linkage is present in many common devices such as transformers, electric motors, and generators. It is also used in technologies like wireless charging and magnetic levitation.

Some common misconceptions about flux linkage include confusing it with magnetic flux, assuming it is constant in a circuit, and thinking it only applies to AC circuits. It is important to understand the difference between magnetic flux and flux linkage, and that it can vary in different parts of a circuit.

## How is flux linkage used in scientific research?

Flux linkage is a crucial concept in understanding electromagnetic induction, which is the basis for many technologies and scientific research. It is used in the study of electromagnetism, electrical engineering, and physics, among other fields.

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