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Purcell says

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 neighborhood of a filamentary current, varies as $1/r$ where $r$ is the distance from the filament, and the integral of $~B \times \textrm{area}~$ diverges as $\int dr /r $ when we extend it down to $r = 0$.

This page clearly shows that the magnetic field through a circular loop of radius $R $ is $\dfrac{\mu _0 I}{2R}$. So the flux through it is $\dfrac{\pi\mu _0 I R^{2}}{2}$, which is obviously finite. Where does the contradiction come from?

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$\dfrac{\mu _0 I}{2R}$ is the magnetic field at the centre of the loop, however, the magnetic field is not the uniform across the plane of the loop, getting stronger closer to the loop and exploding as Purcell notes if the filament radius is zero.

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