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