**Bohr Magneton** is magnetic dipole moment associated with an atom due to orbital motion and spin of an electron.

The **Bohr magneton** (symbol \(\mu_B\)) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by either its orbital or spin angular momentum and is given by:

\(\mu_B = \frac{e\hbar}{2m_e}\)

where,

\(e\) is the elementary charge,

\(\hbar\) is the reduced Planck constant,

\(m_e\) is the electron rest mass,

The value of Bohr magneton in SI units is \(9.27400968(20)×10^{-24}JT^{-1}\)

Index

**History**

**Walther Ritz**(1907) and**Pierre Weiss**are responsible for the concept of elementary magnets. Several researchers suggested that the magneton should have Planck’s constant h even before the Rutherford model of atomic structure.- By postulating that the ratio of electron kinetic energy to orbital frequency should be equal to
*h*, Richard Gans computed a value that was twice as large as the Bohr magneton in September 1911. - In 1911, Romanian physicist
**Stefan Procopiu**discovered the expression for the electron’s magnetic moment. In Romanian scientific literature, the importance is often referred to as the “Bohr–Procopiu magneton.” - In 1911, the
**Weiss magneton**was discovered to be a unit of magnetic moment equal to 1.531024 joules per tesla, or around 20% of the Bohr magneton. - As a result of his atom model, the Danish physicist
**Niels Bohr**obtained the values for the natural units of atomic angular momentum and magnetic moment in the summer of 1913. - Wolfgang Pauli named the it in a 1920 article in which he compared it with the experimentalists’ magneton, which he called the Weiss magneton.

**Bohr Magneton Derivation**

Consider a magnetic dipole moment as a charge q moving in a circle with radius r with speed v.

The current is the charge flow per unit time. Since the circumference of the circle is \(2πr\), and the time for one revolution is \(2πr/v\), one has the current as \(I = qv/2πr\).

The magnitude of the dipole moment is

\(|\mu| = I . \mbox{area} \)

\(|\mu| = (qv/2πr)2πr\)

\(|\mu| = qrp/2m\), where p is the linear momentum.

Since the radial vector r is perpendicular to p,

we have \(\mu = (qr × p)/ 2m = qL /2m\),

where L is the angular momentum.

The magnitude of the orbital magnetic momentum of an electron with orbital-angular momentum quantum l is

\(\mu = e\hbar/2m_e [l(l + 1)]1/2 = \mu_B[l(l + 1)]1/2\)

Here, \(\mu_B\) is a constant called Bohr magneton, and is equal to \(\mu_B = e\hbar/2m_e\)

\(\mu_B\) = (1.6 × 10^{-19}C) × (6.626 × 10^{-34}J · s/2π) /2 × 9.11 × 10^{-31}kg

\(\mu_B\) = 9.274 × 10^{−24}J/T

where T is the magnetic field, Tesla.

For the electron, it is the simplest model possible to the smallest possible current to the smallest possible area closed by the current loop.

**FAQs**

**Does Bohr magneton have any physical significance?**It is a physical constant and the natural unit for describing the magnetic moment of an electron induced by either its orbital or spin angular momentum.

**What are the dimensions of bohr magneton?**Dimensions of bohr magneton are** ***L ^{2} × electric-current*. In SI units, these are the dimensions of the “bohr magneton ” quantity. In electromagnetics, there are other unit systems that can be used to assign various measurements.

**How is Bohr magneton related to the magnetic moment of an electron?**It is the magnitude of the magnetic dipole moment of an orbiting electron with an orbital angular momentum of ħ.