# Off-axis magnetic field strength for permanent magnet calculation/equation

I'm working on a project which uses cylindrical permanent magnets, and I'm trying to determine the expected magnetic field due to these magnets. One problem I run into is that these magnets are usually not completely horizontal above the point at which I want to calculate the expected magnetic field. Assuming the point is on the axis of the cylinder, it's easy enough to calculate the expected magnetic field using the following equation: $$B = \frac{B_r}{2}\left[\frac{D + z}{\sqrt{R^2+(D+z)^2}} - \frac{z}{\sqrt{R^2+z^2}}\right]$$

where $$B_r$$ is the remanence field (found in a magnet's data sheet), $$z$$ is the distance from a face of the cylinder on the axis, $$D$$ is the thickness of the magnet, and $$R$$ is the radius of the cylinder. However, this equation only works on the axis of the cylinder itself. How can I calculate the off-axis magnetic field for a cylindrical permanent magnet?

Note: I don't particularly need the derivation for the off-axis equation for now, although it would definitely be helpful.

• I don't have time for a complete answer right now, but how to work this kind of thing out is covered in problem 5.4 of Jackson's Classical Electrodynamics (third edition). (Problems 5.3 and 5.5 are also related).
– Buzz
Commented Nov 21, 2020 at 2:27
• I'm not sure but this research of multipole cylindrical magnets magnetic field strength expansion off-axis should help. Commented Apr 9, 2021 at 9:26

"Another quite elegant approach (result is derived from the magnetic scalar potential)" Yes, the scalar potential, $$\phi_m$$, easily found from $$B_z=-\partial_z \phi_m$$, can be extended off axis by a Legendre polynomial expansion. This is done on pp. 228-229 of my EM textbook for a current loop, but the same procedure will work for your case.