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Monopole, dipole and multipole

What are monopole, dipole and multipole?

A single charge is called a monopole (from the Greek monos= alone, single). A monopole electric field emanates from it. The magnetic field of a conductor loop or the electric field of 2 oppositely charged particles, on the other hand, is a dipole field (from the Greek di = two). Two opposing charges at a fixed distance correspond to a dipole. There are no magnetic monopoles and therefore only magnets with a north and a south pole. More complicated charge distributions are referred to as multipoles.
Table of Contents
Monopole, dipole, quadrupole and (generally speaking) higher multipoles are designations for correspondingly structured components of electric or magnetic fields. The associated moments, i.e. monopole moment, dipole moment and quadrupole moment, are used to characterise mathematically distinguishable portions of arbitrarily structured electric or magnetic fields. In this regard, the electric field of a point charge is a pure monopole field. This field consists only of a monopole moment.

As a general rule, no monopole for magnetic fields exists. This is expressed by the laws of electromagnetism, i.e. Maxwell's equations. It is said that the lowest non-vanishing multipole moment of the magnetic field is the dipole moment.

Since there are no magnetic monopoles, no permanent magnet with only a single pole can be produced. Every magnet has at least 2 poles, a north pole and a south pole.

The illustration shows an electric monopole. According to Maxwell's equations, it is the source of the electric field. The field lines run away from the charge (towards it in the case of negative charges) like the spines of a hedgehog. An electric dipole is shown in the centre. The magnetic field of a current-carrying conductor loop is shown on the right. A single electron spin, i.e. a so-called elementary magnet, also has this form of magnetic field. Maxwell's equations show that this form of magnetic field is the simplest possible form. From the outside, it resembles the field of an electric dipole. This is why the magnetic field is also referred to as a dipole field. Complicated current distributions also have field components of a higher order. However, there is no magnetic monopole.
The illustration shows an electric monopole. According to Maxwell's equations, it is the source of the electric field. The field lines run away from the charge (towards it in the case of negative charges) like the spines of a hedgehog. An electric dipole is shown in the centre. The magnetic field of a current-carrying conductor loop is shown on the right. A single electron spin, i.e. a so-called elementary magnet, also has this form of magnetic field. Maxwell's equations show that this form of magnetic field is the simplest possible form. From the outside, it resembles the field of an electric dipole. This is why the magnetic field is also referred to as a dipole field. Complicated current distributions also have field components of a higher order. However, there is no magnetic monopole.
The illustration shows the amplitude of electric fields in the plane of the charges. The areas shown are three-dimensional representations of the strength of the electric field in this plane and differ from the direct representations of the fields via field lines as shown in the previous illustration. When field lines are displayed, the direction of the magnetic forces is shown as well. Here on the left, in contrast, you can see a graph for the strength of a monopole electric field, which is displayed using the colour representation of the 3D plot. The electric field is particularly strong at the location of the charge and then diminishes with the square of the distance. An electric dipole field is shown on the right. The dipole field is generated by two opposing charges. Magnetic fields are always dipole fields or fields of a higher order, as there are no single magnetic charges.
The illustration shows the amplitude of electric fields in the plane of the charges. The areas shown are three-dimensional representations of the strength of the electric field in this plane and differ from the direct representations of the fields via field lines as shown in the previous illustration. When field lines are displayed, the direction of the magnetic forces is shown as well.
Here on the left, in contrast, you can see a graph for the strength of a monopole electric field, which is displayed using the colour representation of the 3D plot. The electric field is particularly strong at the location of the charge and then diminishes with the square of the distance. An electric dipole field is shown on the right. The dipole field is generated by two opposing charges. Magnetic fields are always dipole fields or fields of a higher order, as there are no single magnetic charges.

Calculation of the various multipole moments

Mathematically, the calculation of the various multipole moments of any field distribution is solved using the so-called multipole expansion method. This involves a so-called series expansion of the distance dependence for the magnetic field.
In electrodynamics, the movements of electric and magnetic fields give rise to new phenomena such as electromagnetic waves. Here, multipole expansion is also possible. The multipole moments of the radiation fields are then obtained. The lowest non-vanishing multipole radiation is dipole radiation.

We will demonstrate the mathematical method for multipole expansion of magnetic fields of an arbitrary current distribution as an example. The method is very complex and is only presented here to show a typical application of higher mathematics in physics.
The multipole expansion is usually not carried out directly on the formula for the magnetic field or the magnetic flux density, but on the magnetic vector potential \(\vec{A}(\vec{r})\), which depends on the location \(\vec{r}\):

\(\vec{A}(\vec{r})=\frac{\mu_0}{4\pi}\int_{R^3}d^3r^{'}\frac{\vec{j}(\vec{r}^{'} )}{\left|\vec{r}-\vec{r}^{'} \right|}\)
(with the so-called Coulomb calibration \(\vec{\nabla}\cdot\vec{A}(\vec{r})=0\)
Here, \(\vec{j}(\vec{r}^{'} )\) denotes the current distribution at the location of the so-called 'primed' variable \(\vec{r}^{'}\), \(\mu_0\) denotes the magnetic permeability of the vacuum.

\(\left|\vec{r}-\vec{r}^{'} \right|\) denotes the momentary distance between the point at which the magnetic field is determined (\(\vec{r}\)) and the location of the charge distribution (\(\vec{r}^{'}\)).

A Taylor expansion of the function \(\frac{1}{\left|\vec{r}-\vec{r}^{'} \right|}\) by the origin of the primed coordinates (which characterise the current distribution) is now performed:

\(\frac{1}{\left|\vec{r}-\vec{r}^{'} \right|}=\frac{1}{r}+\frac{1}{r^3}\cdot(\vec{r}\cdot\vec{r}^{'})+...\)
Only the first two orders of expansion are shown. The higher orders are abbreviated by ... .
Thus follows:

\(\vec{A}(\vec{r})=\frac{\mu_0}{4\pi\cdot{r}}\int_{R^3}d^3r^{'}\vec{j}(\vec{r}^{'} )+\frac{\mu_0}{4\pi\cdot{r^3}}\int_{R^3}d^3r^{'}\vec{j}(\vec{r}^{'} )\cdot(\vec{r}\cdot\vec{r}^{'})+...\)

With the monopole moment \(\frac{\mu_0}{4\pi\cdot{r}}\int_{R^3}d^3r^{'}\vec{j}(\vec{r}^{'} )\)
and the dipole moment \(\frac{\mu_0}{4\pi\cdot{r^3}}\int_{R^3}d^3r^{'}\vec{j}(\vec{r}^{'} )\cdot(\vec{r}\cdot\vec{r}^{'})\).

The more complicated higher moments are not shown at this point.


Portrait of Dr Franz-Josef Schmitt
Author:
Dr Franz-Josef Schmitt

Dr Franz-Josef Schmitt is a physicist and academic director of the advanced practicum in physics at Martin Luther University Halle-Wittenberg. He worked at the Technical University from 2011-2019, heading various teaching projects and the chemistry project laboratory. His research focus is time-resolved fluorescence spectroscopy in biologically active macromolecules. He is also the Managing Director of Sensoik Technologies GmbH.

The copyright for all content in this compendium (text, photos, illustrations, etc.) remains with the author, Franz-Josef Schmitt. The exclusive rights of use for this work remain with Webcraft GmbH, Switzerland (as the operator of supermagnete.nl). Without the explicit permission of Webcraft GmbH, the contents of this compendium may neither be copied nor used for any other purpose. Suggestions to improve or praise for the quality of the work should be sent via e-mail to [email protected]
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