Finding out the magnetic field at anypoint outside a permanent magnet (Cylinder,Cube)

Question

I am working in a project where I am using different shaped permanent magnets for levitation of diamagnets. I am facing problem while calculating the magnetic field around these permanent magnets. To be specific I am using cylindric magnet and cubic ($\rm NdFeB$) magnets. And I want to find the magnetic field around these shapes of magnet. Only the remanence and the size of the magnet of the are provided. I have come across a formula for cylindrical magnet where it calculates the magnetic field along the symmetry axis of the magnet. But I want to get the magnetic field at any point in space around the cylindrical magnet. Here is the formula:

$$B=\frac{B_r}{2}\left ( \frac{D+z}{\sqrt{{R}^2+{(D+z)}^2}} -\frac{z}{\sqrt{R^2+z^2}}\right )$$

Where $B_r$ is the Remanence field, D is the height of cylinder, R is the radius of cylinder, z is the distance from a pole face on the symmetry axis.

I tried first finding out the scalar potential Φaxis(z) and expanding it to Φ(r, θ) using Legendre polynomial but it doesn’t work. I have already come across $$B_Z,B_X,B_Y (x,y,z)$$ for the cube from one of the paper means now I can easily find $$B(x,y,z)=\sqrt{B_x^2+B_y^2+B_z^2}$$ Can anyone help me to find B(x,y,z) for the cylinder ?

Answer 1

The field of a cylindrical magnet can be viewed as originating from a sheet of current flowing around the perimeter of the bar (the dipole moment/unit volume should be the same as that of the magnet). The field at any point can be approximated by doing a numeric integration (for each component) of the contributions from each little segment of that current. Think of it like a current carrying solenoid. (I'm thinking that this approach could also be used with the cube.)