Magnetic Moment

When you were younger, you may have played with magnets; whether fridge magnets or stronger magnets that can experience an attractive force to one another between materials of varying thicknesses. As a child, the force between the magnets may have seemed like magic! But as you get older, you learn that the attraction is due to the interaction between their respective magnetic fields. If you take two strong magnets and draw a circle with one around the other, you may notice the other magnet begin to spin. As we know, the attractive magnetic force is causing the second magnet to move toward the first magnet, but how exactly do we quantify the rotation it experiences under an external magnetic field? This can be represented using the magnet's magnetic moment. Keep reading to learn more!

Fig. 1 - All magnets have their own magnetic moment to describe how they rotate under an external field.

Magnetic Moment Definition

Firstly, let's define exactly what we mean by a magnetic moment.

The reason why the quantity of a magnetic moment is important in the study of magnets and magnetic dipoles is due to the fact that it allows us to describe the magnetic strength of magnets, not only simple bar magnets and magnetized materials, but also magnetic dipoles generated from current loops or spinning electrons. The magnetic moment is usually measured using the units Ampere-square meters \(\mathrm{A}\,\mathrm{m}^2\), the reason for using units of current and area will become clear when we look at the magnetic moment of a current loop.

Magnetic Moment Formula

A magnet's magnetic moment can be defined by considering placing the magnet in an external applied field. When in an external magnetic field, magnets experience a turning force or torque due to the fact that magnets are always dipoles with opposite polarity at either end.

Using this, the formula used to describe a magnetic moment is given by

\[ \begin{align}\vec{\tau}&= \vec{m} \times \vec{B}\\|\vec{\tau}|&=|\vec{m}|\cdot|\vec{B}|\sin(\theta),\end{align}\]

where \(\vec{\tau}\) is the torque experienced by the magnet measured in units of \(\mathrm{N}\,\mathrm{m}\), \(\vec{m}\) is the magnetic moment measured in units of \(\mathrm{A}\,\mathrm{m^2}\), and \(\vec{B}\) is the magnetic field vector measured in units of teslas \(\mathrm{T}\). This formula shows us that the magnetic moment is a vector quantity, this vector is taken to point from the south to the north pole of the magnet.

Let's consider an example where we use the magnetic moment equation.

Magnetic Dipole Moment

We often think of magnets in terms of magnetic dipoles, that is, the north and south poles of a magnet separated by a short distance. These separate poles are only hypothetical, as no magnetic monopole has ever been observed in nature. So, the magnetic dipole model is a hypothetical model we can use to understand the magnetic moment. As we have already seen, we can use this dipole to define the direction of the magnetic moment.

Fig. 2 - A bar magnet is a type of dipole where the magnetic field lines leave the north pole and end at the south pole.

This model allows us to understand why magnetic fields produce torques on a magnetic dipole, as the opposite poles experience opposing forces. These opposing forces produce the torque experienced by the magnet, which is where we get the name magnetic moment, as in mechanical moments produced by forces acting some perpendicular distance from a pivot point. In this theoretical model, the magnetic moment is given by

\[\vec{m}=p\vec{l}\]where \(p\) is the fictitious monopole strength and \(\vec{l}\) is the vector in between the two monopoles.

Magnetic Moment of Current Loop

Magnetic moments are not only found in typical magnets such as bar magnets and magnetized ferromagnetic materials, but also in current loops. This was first discovered by Hans Christian Ørsted in 1820. When a current flows in a closed loop, it induces a magnetic field that flows through the surface area enclosed by the loop. This means that a small loop of current is in effect a magnetic dipole, and we can define its magnetic dipole moment.

The magnetic moment of a current loop is defined by the current flowing through the loop \(I\) and the area enclosed by the loop \(\vec{A}\):

\[\vec{m}=I\vec{A}.\]

The area is defined to be a vector, with the magnitude being the surface area with the vector directed perpendicular to the surface. The choice of direction for the area vector and hence the magnetic moment vector is determined using the right-hand rule.

Magnetic Moment of Electron

The magnetic moment is, in fact, an intrinsic quantity of fundamental particles like the electron. This arises due to the quantum mechanical property of electrons known as spin. This spin refers to the intrinsic angular momentum of particles about their own axis. It's not quite correct to think of the electrons as actually spinning about some axis, however it's a useful picture to have in your head.

Because electrons have charge, their intrinsic spin produces a magnetic moment, which can be thought to arise from this movement of charge similar to the way that a current produces a magnetic moment. The magnetic moment of an electron is a fundamental quantity, known as the Bohr Magneton:\[\mu_B=\frac{e\hbar}{2m_e}=9.27\,\mathrm{J}\,\mathrm{T}^{-1}.\]