I have a magnetic disk (Radius R, height h) that creates magnetic field lines (axisymmetrical). I simulated the field lines, exported the data and am now trying to fit a function into the data to have a analytical solution for the magnetic flux density of my specific magnet. For several r's fixed I have the magnetic flux Density $B_z(z)$ as well as $B_r(z)$
I need functions of the following form: $B_z(r,z)$ and $B_r(r,z)$ since I think $B_\phi (r,z)$ is $0$. So I am wondering how I need to start. I found the following equation online but I think its only for the absolute value, not $B_z$ and $B_r$ how I need it:
$$B(r,z) = \frac{\mu_0 m}{4\pi (z^2+r^2)^{\frac{3}{2}}}\sqrt{1+\frac{3z^2}{z^2+r^2}}$$
Can someone help me how my function needs to look? I was thinking something like:
$$B_z(r,z) = \frac{a}{b+((z+c)^2+(r+d)^2)^{\frac{3}{2}}}\sqrt{1+\frac{3z^2}{z^2+r^2}}$$
or am I completely off now? And furthermore, how would I change to cartesian coordinates then?
Many thanks for your help!
Best Answer
My answer will probably be a little off topic, but why do you need this kind of analytical function ? Since you have the magnetic flux density (FEM simulation I guess), you can use any interpolation to get the B value anywhere... Unless you have a very specific need, if you only want to get the value of B anywhere, that is probably the easiest solution. I would use, for instance, scipy.interpolate.griddata or even interp2d if your mesh is regular.
From an academic point of view, the analytical solution of the induced magnetic field of a coil/magnet does not exist without the use of simplifications (ex. being far for the magnet (your equation), near the axe...).