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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) = mu

_{0}*I/(2*pi*r) - this bit I get. However, the total flux linkage is then given by:

phi = Int[B dA] = Int

_{a}

^{b}[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 = mu

_{0}*I*r/(2*pi*R

^{2}), 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

_{r}

^{R}[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.

Thanks for your help.