This is quoted form Edward Purcell's Electricity and Magnetism; it tells about the problem of computing magnetic flux during self-induction for a wire with zero-diameter:
You may think that one of the rings we considered earlier would have made a simpler example to illustrate the calculation of self-inductance. However, if we try to calculate the inductance of a simple circular loop of wire, we encounter a puzzling difficulty. It seems a good idea to simplify the problem by assuming that the wire has zero diameter. But we soon discover that, if finite current flows in a filament of zero diameter, the flux threading a loop made of such a filament is infinite! The reason is that the field $B$, in the neighbourhood of a filamentary current, varies as $1/r,$ where $r$ is the distance from the filament, and the integral of $B \times \text{area}$ diverges as $\int (dr/r)$ when we extend it down to $r= 0$. To avoid this, we may let the radius of the wire be finite, and not zero, which is more realistic anyway. This may make the calculation a bit more complicated, but that won't worry us. The real difficulty is that different parts of the wire now appear as different circuits, linked by different amounts of flux. We are no longer sure what we mean by the flux through the circuit. In fact, because the electromotive force is different from the different filamentary loops into which the circuit can be divided, some redistribution of current density must occur when rapidly changing currents flow in the ring. Hence the inductance of the circuit may depend somewhat on the rapidity of $I$ ...
We avoided this embarrassment in the toroidal coil example by ignoring the field in the immediate vicinity of the individual turns of the winding. Most of the flux does not pass through the wire themselves, and whenever that is the case the effect we have just been worrying about will be unimportant.
Why did he take $r= 0$ in the integral? Did he want to find the flux of the field on the wire also? Also, even if he takes a non-zero diameter wire, shouldn't there be magnetic field on the wire then also? How did by taking non-zero diameter wire, problem with $r= 0$ in the integration vanished? Shouldn't there be magnetic field lines passing through the wire even if the wire is having non-zero diameter?
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