# Magnetic field path on a surface with axial symmetry

$\newcommand{\dv}[2]{\frac{\mathrm{d} #1}{\mathrm{d}#2}}$ I'm trying to find the path a magnetic field line takes along an axially symmetric magnetic surface.

I'm given: - $r,z$ points along a poloidal contour (ie a slice at one toroidal position) - $B_r$, $B_z$, $B_\phi$ at each one of these points - the number of times it must wrap around poloidally for each time it goes around toroidally - It wraps back onto itself (ie the last point is the first point)

So basically I need the $\phi$ points for each one of these $r,z$ points. So I know that $$\dv{r}{z}= \frac{B_r}{B_z}$$, but how would that work for phi because $\dv{\phi}{r}$ does not equal $\frac{B_\phi}{B_r}$.

The answer for small $\mbox{d}r$ and $\mbox{d}\phi$ is to use $$\frac{r\mbox{d}\phi}{\mbox{d}r} = \frac{B_\phi}{B_r}$$