# Multipole magnet

Multipole magnets are magnets built from multiple individual magnets, typically used to control beams of charged particles. Each type of magnet serves a particular purpose.

## Magnetic field equations

The magnetic field of an ideal multipole magnet in an accelerator is typically modeled as having no (or a constant) component parallel to the nominal beam direction ($z$  direction) and the transverse components can be written as complex numbers:

$B_{y}+iB_{x}=C_{n}\cdot (x+iy)^{n-1}$

where $x$  and $y$  are the coordinates in the plane transverse to the nominal beam direction. $C_{n}$  is a complex number specifying the orientation and strength of the magnetic field. $B_{x}$  and $B_{y}$  are the components of the magnetic field in the corresponding directions. Fields with a real $C_{n}$  are called 'normal' while fields with $C_{n}$  purely imaginary are called 'skewed'.

## Stored energy equation

For an electromagnet with a cylindrical bore, producing a pure multipole field of order $n$ , the stored magnetic energy is:

$U_{n}={\frac {n!^{2}}{2n}}\pi \mu _{0}\ell N^{2}I^{2}.$

Here, $\mu _{0}$  is the permeability of free space, $\ell$  is the effective length of the magnet (the length of the magnet, including the fringing fields), $N$  is the number of turns in one of the coils (such that the entire device has $2nN$  turns), and $I$  is the current flowing in the coils. Formulating the energy in terms of $NI$  can be useful, since the magnitude of the field and the bore radius do not need to be measured.

Note that for a non-electromagnet, this equation still holds if the magnetic excitation can be expressed in units Amperes.

### Derivation

The equation for stored energy in an arbitrary magnetic field is:

$U={\frac {1}{2}}\int \left({\frac {B^{2}}{\mu _{0}}}\right)\,d\tau .$

Here, $\mu _{0}$  is the permeability of free space, $B$  is the magnitude of the field, and $d\tau$  is an infinitesimal element of volume. Now for an electromagnet with a cylindrical bore of radius $R$ , producing a pure multipole field of order $n$ , this integral becomes:

$U_{n}={\frac {1}{2\mu _{0}}}\int ^{\ell }\int _{0}^{R}\int _{0}^{2\pi }B^{2}\,d\tau .$

Ampere's Law for multipole electromagnets gives the field within the bore as:

$B(r)={\frac {n!\mu _{0}NI}{R^{n}}}r^{n-1}.$

Here, $r$  is the radial coordinate. It can be seen that along $r$  the field of a dipole is constant, the field of a quadrupole magnet is linearly increasing (i.e. has a constant gradient), and the field of a sextupole magnet is parabolically increasing (i.e. has a constant second derivative). Substituting this equation into the previous equation for $U_{n}$  gives:

$U_{n}={\frac {1}{2\mu _{0}}}\int ^{\ell }\int _{0}^{R}\int _{0}^{2\pi }\left({\frac {n!\mu _{0}NI}{R^{n}}}r^{n-1}\right)^{2}\,d\tau ,$

$U_{n}={\frac {1}{2\mu _{0}}}\int _{0}^{R}\left({\frac {n!\mu _{0}NI}{R^{n}}}r^{n-1}\right)^{2}(2\pi \ell r\,dr),$

$U_{n}={\frac {\pi \mu _{0}\ell n!^{2}N^{2}I^{2}}{R^{2n}}}\int _{0}^{R}r^{2n-1}\,dr,$

$U_{n}={\frac {\pi \mu _{0}\ell n!^{2}N^{2}I^{2}}{R^{2n}}}\left({\frac {R^{2n}}{2n}}\right),$

$U_{n}={\frac {n!^{2}}{2n}}\pi \mu _{0}\ell N^{2}I^{2}.$