I am trying to calculate the total magnetic flux through the surface of a current loop. I feel that this flux should be finite and nonzero -- so far any attempt in calculating failed. Can this really not be done?
For example: We know \begin{equation}d\textbf{B}=\dfrac{\mu_0}{4\pi}\,\dfrac{I\,d\textbf{l}\times\hat{\textbf{R}}}{|\textbf{R}|^2}\end{equation} where $d\textbf{l}=\hat{\boldsymbol{\phi}}\cdot a\,d\phi'$, for a loop of radius $a$ centered around the origin and lying in the $z=0$ plane in cylindrical coordinates. Hats denote unit vectors. The distance vector is $\textbf{R}=\textbf{r}-\textbf{r}'$ and $|\textbf{R}|=\sqrt{r^2+a^2-2ar\cos(\phi-\phi')}$.
Thus, the magnetic flux density is only $z$-directed: $$dB_z=\dfrac{\mu_0 I}{4\pi}\,\dfrac{a}{|\textbf{R}|^2}\,d\phi'$$
The total flux is simply the surface integral, \begin{multline}\Phi=\int\hspace{-0.5em}\int\textbf{B}\cdot d\textbf{s}=\int\hspace{-0.5em}\int B_z\,r\,d\phi\,dr=\int\hspace{-0.5em}\int\hspace{-0.5em}\int dB_z\,r\,d\phi\,dr\\=\dfrac{\mu_0 Ia}{4\pi}\int_0^a\hspace{-0.5em}\int_0^{2\pi}\hspace{-0.5em}\int_0^{2\pi}\dfrac{r}{r^2+a^2-2ar\cos(\phi-\phi')}\,d\phi'\,d\phi\,dr\end{multline}
However, this does not converge, as far as I can tell. Similar issues arise with formulations via magnetic vector potential or current density $\textbf{J}$ instead of current $I$.
Via inductance, we know that $\Phi=LI$ and that in this theoretical example, the inductance is purely a geometric problem. However... $L\rightarrow\infty$ doesn't make sense to me.
Can anybody help me out and point me into the right direction here?