# 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 with the here stated formulas on the axis north-south-pole. Alternatively, computer programs can calculate fields in the whole room.
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 B field strength with a direction 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 justifying force is proportional to the strength of the B field.
There are no simple formulas that calculate this field for the various magnetic shapes. Computer programs were developed for that purpose (see below). There are simple formulas for less complex symmetrical geometries, which indicate the B field on a symmetry axis in north-south pole direction. Subsequently, we are glad to share these with you.

## Formula for block magnet flux density

Formula for the B field on the symmetry axis of an axially magnetised block magnet (block or cube):
\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.

## Calculation table for flux density calculation

The formulas above can also be calculated in a table. Enter magnet details in the yellow fields, which automatically calculates the flux density. The following versions are available:

## B fields in the whole room

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.
Just like other tools, FEMM calculates and charts only half of the magnet because the B fields are symmetrical. The other half you have to imagine mirrored on the left.