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 dψ1 = 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?