# How do you calculate the magnetic flux density?

The magnetic flux density is also called "B field" or "magnetic induction". The B field of our super magnets can be calculated on the north-south pole axis using the formulas given here. Additionally, we also provide you with tables (Excel/OpenOffice) you can use to automatically calculate the magnetic flux density. By contrast, calculating the B fields of an entire space is much more complex and requires the use of computer programs.
The magnetic flux density of a magnet is also called "B field" or "magnetic induction". It is measured in tesla (SI unit) or gauss (10 000 gauss = 1 tesla).
A permanent magnet produces a B field in its core and in its external surroundings. A directional B field strength can be attributed to each point within and outside of the magnet. If you position a small compass needle in the B field of a magnet, it orients itself toward the field direction. The effecting force is proportional to the strength of the B field.
There are no simple formulas for calculating the magnetic flux density of the various magnetic shapes. Computer programs were developed for that purpose (see below). However, simple formulas do exist for less complex symmetrical geometries, enabling you to calculate the B field on a symmetry axis in north-south pole direction. We are happy to share these formulas for calculating the magnetic flux density with you below.

## Formula for block magnet flux density

Formula for the B field on the symmetry axis of an axially magnetised block or cube magnet:
\begin{aligned}B &= \frac{B_r}{\pi}\left[arctan\bigg(\frac{LW}{2z\sqrt{4z^2+L^2+W^2}}\bigg)- arctan\bigg(\frac{LW}{2(D+z)\sqrt{4(D+z)^2+L^2+W^2}}\bigg)\right]\end{aligned}
Br: Remanence field, independent of the magnet's geometry (see physical magnet data)
z: Distance from a pole face on the symmetry axis
L: Length of the block
W: Width of the block
D: Thickness (or height) of the block
The unit of length can be selected arbitrarily, as long as it is the same for all lengths.

## Formula for cylinder magnet flux density

Formula for the B field on the symmetry axis of an axially magnetised cylinder magnet (disc or rod):
\begin{aligned}B &= \frac{B_r}{2}\left(\frac{D+z}{\sqrt{R^2+(D+z)^2}}-\frac{z}{\sqrt{R^2+z^2}}\right)\end{aligned}
Br: Remanence field, independent of the magnet's geometry (see physical magnet data)
z: Distance from a pole face on the symmetrical axis
D: Thickness (or height) of the cylinder
R: Semi-diameter (radius) of the cylinder
The unit of length can be selected arbitrarily, as long as it is the same for all lengths.

## Formula for ring magnet flux density

Formula for the B field on the symmetry axis of an axially magnetised ring magnet:
\begin{aligned}B &= \frac{B_r}{2}\left[\frac{D+z}{\sqrt{R_a^2+(D+z)^2}}-\frac{z}{\sqrt{R_a^2+z^2}}-\left(\frac{D+z}{\sqrt{R_i^2+(D+z)^2}}-\frac{z}{\sqrt{R_i^2+z^2}}\right)\right]\end{aligned}
Br: Remanence field, independent of the magnet's geometry (see physical magnet data)
z: Distance from a pole face on the symmetry axis
D: Thickness (or height) of the ring
Ra: Outside radius of the ring
Ri: Inside radius of the ring
The unit of length can be selected arbitrarily, as long as it is the same for all lengths.
The formula for ring magnets shows that the B field for a ring magnet is composed of the field of a larger cylinder magnet with the radius Ra minus the field of a smaller cylinder magnet with the radius Ri.

## Formula for sphere magnet flux density

Formula for the B field on the symmetry axis of an axially magnetised sphere magnet:
\begin{aligned}B &= B_r\frac{2}{3}\frac{R^3}{(R+z)^3}\end{aligned}

Br: Remanence field, independent of the magnet's geometry (see physical magnet data)
z: Distance from the sphere edge on the symmetry axis
R: Semi-diameter (radius) of the sphere
The unit of length can be selected arbitrarily, as long as it is the same for all lengths.

## Table with formulas for flux density calculation

The above-mentioned flux density formulas can also be conveniently calculated in a table. Enter the magnet details in the yellow fields and the flux density will be automatically calculated. The following versions are available:

Source for the above-mentioned formulas: Article at researchgate.net

## Calculating the B fields of an entire space

For calculating the B fields aside from symmetry axes or the fields of various magnetic shapes, there are very elaborate and often very expensive computer programs, which can calculate B fields and much more.
A free software that is limited to rotation-symmetric magnets is FEMM ("Finite Element Method Magnetics").
Just like other tools, FEMM calculates and charts only one half of a magnet because the B fields are symmetrical. You have to imagine the other half mirrored on the left.
B field of half a magnet (disc magnet), illustrated with FEMM