# Magnetic Moment

In Electrostatics, electric monopoles (charges) exist independently but in Magnetostatics, there is no analog of electric charges since magnetic monopoles do not exist (according to Gauss’ law). The magnetic moment of an object is a vector quantity, which describes the magnetic field created by that object. When a magnet is exposed to an external magnetic field, it tries to align with the field. The magnet, having a magnetic moment, experiences a torque. The magnetic moment is defined in terms of this torque. Magnets, current-carrying loop, atoms, and molecules, subatomic particles, etc. have magnetic moments.

### Motivation

Magnetic moment can be defined by drawing an analogy from electrostatics. If two equal and opposite electric charges are separated by a small distance, they constitute an electric dipole. An electric dipole of dipole moment pE experiences a torque NE  in an external electric field E.

$N_{E} = p_{E}\times E$

Similar behavior can be seen in a current-carrying loop. The torque N experienced by a loop carrying current I, in a uniform magnetic field  B is given by,

N = IA x B

Here, A is a vector having a magnitude equal to the area enclosed by the loop and direction, normal to the area.

The quantity IA plays the role of electric dipole moment in magnetostatics. It is defined as the magnetic moment (or magnetic dipole moment) of the loop.

m = IA

It can be defined from the magnetic moment formula,

N=m x B

A current-carrying loop can be considered as a magnet having two poles. The following objects have magnetic moments,

• Permanent magnet

• Electromagnet

• Current carrying loop

• Atoms and molecules

• Subatomic particles (spin magnetic moment)

### Dimension and Unit of Magnetic Moment

Magnetic moment is a vector quantity having dimension [IL2]. The SI unit of magnetic dipole moment is Am2 . The CGS unit is emucm2. These two units are related as,

1 emucm2 = 10-3 Am2

### Pole Strength of a Magnet

Magnetic monopoles do not exist however to draw an analogy from electrostatics, a concept of “pole strength” can be defined, which is equivalent to electric charge. If a bar magnet of length r is inclined at an angle θ with the direction of a uniform magnetic field B, two equal and opposite forces act on its two ends. The magnet tends to align with the magnetic field. The force on the north pole points towards B and the force on the south pole is opposite to that. Both of the forces have magnitude F

This couple of force exerts a torque on the magnet, due to which it rotates. The moment of a couple (i.e. torque) is given by,

N = F × r

The direction of the torque is perpendicular to the plane of the paper and its magnitude is given by,

N = F r sinθ

Considering the magnetic moment of the bar magnet to be m, the torque has a magnitude,

N = mBsinθ

Comparing the expressions,

F r sinθ  = mBsinθ

$F = \frac{m}{r} B$

The quantity $q_{m} = \frac{m}{r}$ is equivalent to electric charge and it is referred to as the “pole strength”. the strength of north pole is taken to be + $q_{m}$ and that of south pole is chosen as - $q_{m}$.

According to the pole strength formula, the pole strength of a magnet is given by the ratio of magnetic moment to its effective length (called the magnetic length). SI unit of pole strength is A . m.

### Force and Potential Energy

The force on a magnetic moment m due to a magnetic field B is given by,

F = (m . ▽)B

The potential energy is as follows,

U = -m . B

### Magnetic Moment in Chemistry

An electron revolving around the nucleus of an atom, constitutes a closed current-carrying loop. The magnetic moment of an electron is,

m = $-\frac{\mu_{g}}{h} L$

Here, L is the angular momentum and it is quantized in units of Planck’s constant ħ. $\mu_{B}$ is called Bohr magneton defined as,

$\mu_{B} = \frac{e\hbar}{2m_{e}}$

Here, the mass of electron $m_{e} = 9.1\times 10^{-31}$ kg

Charge of electron $e = 1.6\times 10^{-19}$ C

Planck’s constant $h = 2\pi\hbar = 6.626\times 10^{-34}$ J.s.

The magnetic moment due to the orbital motion of an electron (with orbital quantum number l) as a magnitude,

$m_{l} = \sqrt{l(l + 1)}\mu_{B}$

Apart from this, an electron has magnetic moment due to its intrinsic spin ($s=\frac{1}{2}$). It has a magnitude given by the spin magnetic moment formula,

$m_{s} = 2\sqrt{s(s + 1)}\mu_{B}$

Protons and neutrons are spin half $(s = \frac{1}{2})$ particles. The magnetic moments have magnitudes,

$m_{p} = g_{p}\sqrt{s(s + 1)} \frac{e\hbar}{2M_{p}}$

$m_{n} = g_{n}\sqrt{s(s + 1)} \frac{e\hbar}{2M_{n}}$

Here, $M_{p}$ and $M_{n}$ are the masses of proton and neutron respectively. $g_{p}$ and $g_{n}$ are empirical constants.

Atoms consist of multiple electrons. The total magnetic moment of an atom is the resultant of the magnetic moments of individual electrons. It depends on the number of unpaired electrons present in an atom. The total magnetic moment has a magnitude in the units of Bohr magneton,

$m = \sqrt{n(n + 2)}$

Here, n is the number of unpaired electrons. For example, the magnetic moment of Ni2+ is 2.45 Bohr magneton since it has 2 unpaired electrons.

### Did You Know?

• Materials, consisting of atoms with unpaired electrons are called paramagnetic. Each atom behaves as a tiny dipole moment. Normally, they remain randomly oriented. But in the presence of an external magnetic field, the dipoles tend to get arranged parallel to the magnetic field.

• Total magnetic moment of an electron is a sum of orbital and spin magnetic moments.

• Magnetic moment of an electron is opposite to its angular momentum. Like angular momentum and spin, magnetic moment is also quantized.

• Ferromagnetic atoms have a much higher value of magnetic moment than that of paramagnetic atoms.

• Most of the paramagnetic materials are colored.

• Early theories concerning magnetostatics considered the existence of magnetic monopoles. But Gauss’ law discards the concept of monopoles.

• Magnetic field strength at a point, due to a magnetic moment, is inversely proportional to the cube of the distance of that point from the dipole.