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?
Best Answer
The trouble arises, I believe, because you're considering the field to be due to a current in a wire of zero thickness, so the flux density approaches infinity as you approach the wire, and this makes the flux integral blow up. If you consider current spread over a finite cross-sectional area of wire this problem goes away. There are other mathematical difficulties, of course, but they can be handled by approximation methods, and you'll find formulae for flux due to a circular loop on the internet.