Magnetic Moment: Formula, Magnetic Properties & Bohr Magneton

Magnetic moment refers to the strength and orientation of a magnet or any other object that produces magnetic field. It can also be defined as the magnetic dipole moment of an object. The ability of a magnetic dipole to align itself with the external magnetic field can be measured by the magnetic dipole moment. Permanent magnets, electromagnets, some molecules, electrons, astronomical objects like planets, stars etc. are some of the examples of the objects having magnetic moment. In case of electrons, magnetic moment is caused by its inherent spin and electric charge properties. 

Magnetic moment is a vector quantity. The direction of the magnetic moment is from the south pole to the north pole of a magnet. The magnetic field produced by a magnet is directly proportional to the magnetic moment. 

Check Also: NCERT Solutions for Class 12 Physics Chapter 5 Magnetism and Matter

Key Terms: Magnetic Moment, Magnetic Dipole, Magnetic Fields, Electron, Torque, Bohr Magneton, Magnetometer, Electric Field

What is Magnetic Moment?

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A substance that may repel or attract other substances is referred to as a "magnetic material." The arrangement of electrons, also known as the magnetic moment of the material, determines whether these materials are attracted to or repel one another. The magnetic dipole moment of an object is a measure of the torque experienced by it in the presence of a magnetic field. 

Magnetic Moment Electron

Magnetic Properties of Solids

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The magnetic characteristics of the ions or atoms that make up a solid determine the magnetic properties of that solid. The magnetism and magnetization of a material can be impacted by the mobility of electrons within an atom. Because each electron in an atom behaves like a magnet, as a result the entire solid has magnetic properties. An atom's magnetic activity is influenced by the ways in which electrons flow within it. They move in one of the two different ways: 

• Electrons revolving around its nucleus.
• Spinning of electrons along their own axis; the + and - signs denote the direction of the spin. 

Because of these constant movements, the electrons are surrounded by an electric field that resembles a current loop and thus generates magnetic moment. Based on their magnetic properties, solids are divided into five classes:

Paramagnetic

These compounds exhibit weak magnetization in the presence of an external magnetic field. Such substances are weakly attracted by magnetic fields. If a paramagnetic substance is taken out of the magnetic field, it loses its magnetic properties. Paramagnetism is created when the magnetic field magnetizes at least one pair of mismatched electrons in their orbit shell. Cu²⁺, O₂ are some of the examples. 

Diamagnetic

Diamagnetic substances are the substances that are weakly repelled by the magnetic fields. They do so because the magnetic property that has settled inside of them runs counter to the magnetic fields. In diamagnetic substances, all of the electrons in the orbital are completely filled and are paired. This is why their atoms have zero magnetic moments. Diamagnetic substances behave like an insulator. Examples include NaCl, Benzene etc. 

Ferromagnetic

Ferromagnetic substances are strongly attracted by the magnetic fields. These solids become permanently magnetized when they are subjected to an external magnetic field. It means they can maintain their magnetic properties even in the absence of external magnetic fields. Ferromagnetic substances have distinct groups of metal ions known as "domains." Every domain is comparable to a little magnet. In an electromagnetic environment, these domains organize and align themselves with the magnetic field. All the electrons in these substances are unpaired and aligned in the same direction. Examples of ferromagnetic solids include cobalt, nickel etc. 

Antiferromagnetic

These substances have a net zero magnetic moment. This is because their dipole moments are arranged in a compensatory manner. The examples of such substances include MnO, V₂O₃ etc. 

Ferrimagnetic

These substances emerge when magnetic moments are aligned in both directions (parallel and antiparallel), but in unequal proportions. They have a small net magnetic moment. Some materials might entirely lose their ferromagnetism upon heating. Examples include ferrites made of magnesium, zinc, and iron.

Magnetic Moment Formula

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The magnetic moment measures the torque acting on an object when it is placed in a magnetic field. This relation can be expressed by the following equation;

\(\tau = m \times B\)

where,

• \(\tau\) is the torque experienced by the object
• m is the magnetic moment
• B is the magnetic field 

Derivation of the Magnetic Moment Formula

Let us consider an object placed in a magnetic field B, 

force on the N-pole = mB (in the direction of the magnetic field)

force on the S-pole = mB (opposite to the direction of the magnetic field)

The forces acting on the north and the south poles are equal in magnitude but opposite in direction. They create a parallel couple rotating the magnet clockwise and producing a net torque on the magnet. Thus, the torque can be written as, 

\(\tau\) = moment of the couple

\(\tau\) = mB × 2l sin \(\theta\)         (\(\theta\) is the angle between the magnetic axis of the magnet and the magnetic field)

Now, we know that

M = m × 2l

Thus, \(\tau\) = MB sin\(\theta\) 

Or, \(\tau\) = M × B

Bohr Magneton

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Natural magnetic dipoles make up the majority of elementary particles. For instance, the electron functions as a magnetic dipole and has a spin magnetic dipole moment. Since the electron is a point entity and lacks both an area A and the ability to revolve about itself, its existence depends on this magnetic moment.

According to Neil Bohr's atom model, the negatively charged electron orbits a positively charged nucleus in a circle with radius r. A revolving electron in a small channel makes up an electric current. An ordinary current flows clockwise as a result of the electron's anticlockwise motion.

Current is given as,

i = e ⁄ T

where, e is the charge of an electron and T is the period of the electron’s revolution.

If v is the electron’s orbital velocity, then

T = 2πr ⁄ v

where, r is the radius of the orbit

Therefore, i = ev ⁄ 2πr

There will be an orbital magnetic moment μl due to the electron’s orbital motion,

μl = i A

Here, A is the area of the orbit

A = πr²

μl = (ev ⁄ 2πr) (πr²)

μl = evr ⁄ 2

Consider the mass of the electron be m,

μl = (e ⁄ 2m) (mvr)

The electron’s angular momentum (l) around the central nucleus is (mvr),

μl = (e ⁄ 2m)l

(μl ⁄ l = e ⁄ 2 m is called gyromagnetic ratio and is a constant)

Its value is 8.8 × 10¹⁰ C kg^-1. The angular momentum, according to Bohr, has only a discrete set of values given by the equation,

l = nh ⁄ 2π

Here, n is a natural number and h is Planck’s constant.

Substitute the formula of l into the formula of μl,

μl = (e ⁄ 2m) (nh ⁄ 2π)

= neh ⁄ 4πm

The minimum value of the magnetic moment is obtained at n = 1.

μl = eh ⁄ 4πm

It is called Bohr Magneton.

By entering the values of e, h, and m, it is discovered that the Bohr magneton's value is 9.27 × 10^-27 J/T.

In addition to the magnetic moment caused by orbital motion, the electron also has a magnetic moment due to its spin. The vector sum of an electron's magnetic orbital and spin moment is the resultant magnetic moment.

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Things to Remember

• Magnetic moment is a measure of the orientation and the strength of the magnet producing magnetic field. 
• The formula for magnetic moment is given by \(\tau\) = M × B. 
• The magnetic moment of an electron, or electron magnetic dipole moment, is caused by its inherent spin and electric charge properties.
• The magnetic moment is generated by the motion of electrons – revolution around the nucleus and spinning on its own axis. 
• The magnetic characteristics of the ions or atoms that make up a solid determine the magnetic properties of the solid.