What is Bohr magneton?: An Introduction
In 1913, values of angular momentum and magnetic moment were obtained by Bohr. In 1920, Wolfgang Pauli gave it the name Bohr Magneton. Bohr magneton is a physical constant value and it is the magnetic moment of an electron due to its angular momentum and also orbital momentum. Its value is 9.274 x 10-24 JT-1.
What is a Magnetic Moment?
Magnetic moment is also sometimes known as magnetic dipole moment. Magnetic moment is the measure of the tendency of an object to align itself with a magnetic field. It can also be stated as the magnetic strength and orientation of an object that produces a magnetic field. Magnetic moment is measured using a device called a Magnetometer. Magnetic moment is a vector quantity. S.I. unit is Am2. It can be given by formula \[\tau = m \times B\]. Here, \[\tau \] is Torque, m is magnetic moment and B is the external magnetic field.
Dipole moment of the electron
What is Bohr Magneton?
Bohr magneton is a physical constant value. Bohr magneton is the magnetic moment of the electron due to its angular momentum and also its orbital momentum.
Bohr Magneton Formula
Electrons move in a circular path with a velocity v and path of radius r. The time is given by equation:
\[T = \dfrac{{2\pi r}}{\upsilon }\],
Because the electron is moving, there is some charge developed which we can give by equation:
\[I = \dfrac{{ - e\upsilon }}{{2\pi r}}\],
The magnetic moment can be given by equation, \[m = IA\],
Here A is the area of the circular path.
\[\mu = IA\]
\[\mu = \dfrac{{ - e\upsilon }}{{2\pi r}}\pi {r^2}\]
\[\mu = \dfrac{{ - evr}}{2}\]
Multiplying and dividing above equation by mass of electron
\[\mu = \dfrac{{ - evr}}{{2{m_e}}}{m_e}\]
\[{m_e}\upsilon r = L\], Which is angular momentum
\[\therefore \mu = \dfrac{{ - e}}{{2{m_e}}}\dfrac{{nh}}{{2\pi }}\] since, \[L = \dfrac{{nh}}{{2\pi }}\]
\[\mu = \dfrac{{neh}}{{4\pi {m_e}}}\]
Here Bohr Magneton is, \[{\mu _B} = \dfrac{{eh}}{{4\pi {m_e}}}\]
We can write \[\dfrac{h}{{2\pi }} = \hbar \]
\[\therefore \] Bohr magneton formula is \[{\mu _B} = \dfrac{{e\hbar }}{{2\pi }}\]
From this formula Bohr magneton value is 9.274 x 10-24 J/T
And SI unit of Bohr magneton is J/T (Joules per Tesla)
Interesting Facts
The strongest magnet in the universe is a star and it is called magnetar.
Our planet earth is like one big bar magnet
Magnets always have two poles
Solved Problems
1. What is the ratio of Bohr magneton to the nuclear magneton?Ans: Bohr magneton is a physical constant value and it is the magnetic moment of an electron due to its angular momentum and also orbital momentum. Nuclear magneton is the magnetic dipole moment of other heavier particles like protons, nuclei, etc. Bohr magneton is given by formula \[{\mu _B} = \dfrac{{e\hbar }}{{2{m_e}}}\]. Nuclear magneton is given by formula \[{\mu _N} = \dfrac{{e\hbar }}{{2{m_p}}}\]. Taking the ratio of these two we get, \[\dfrac{{{\mu _B}}}{{{\mu _N}}} = \dfrac{{{m_p}}}{{{m_e}}}\].
Ans: Electron moves in a circular path with a velocity v and path of radius r. The time is given by equation,
\[T = \dfrac{{2\pi r}}{\upsilon }\]
Because the electron is moving, there is some charge developed which we can give by equation,
\[I = \dfrac{{ - e\upsilon }}{{2\pi r}}\]
The magnetic moment can be given by equation, \[{M_0} = IA\],
Here A is the area of the circular path.
\[{M_0} = IA\]
\[{M_0} = \dfrac{{ - e\upsilon }}{{2\pi r}}\pi {r^2}\]
\[{M_0} = \dfrac{{ - evr}}{2}\]
Multiplying and dividing above equation by mass of electron
\[{M_0} = \dfrac{{ - evr}}{{2{m_e}}}{m_e}\]
\[{M_0} = \dfrac{{ - e}}{{2{m_e}}}{m_e}\upsilon r\]
We know that, angular momentum, \[{m_e}\upsilon r = {L_0}\]
Gyromagnetic ratio=magnetic moment/angular momentum
Gyromagnetic ratio=\[\dfrac{{{M_0}}}{{{L_0}}} = \dfrac{e}{{2{m_e}}}\]
Summary
Bohr magneton is a physical constant value. Bohr magneton is the magnetic moment of the electron due to its angular momentum and also its orbital momentum. The Bohr magneton formula is \[{\mu _B} = \dfrac{{e\hbar }}{{2\pi }}\]. Bohr magneton value is 9.274 x 10-24 J/T. SI unit of Bohr magneton is J/T (Joules per Tesla).
FAQs on Bohr Magneton: A Detailed Summary
1. What is the Bohr magneton and what is its physical significance in physics?
The Bohr magneton (μB) represents the smallest possible magnetic dipole moment associated with an electron due to its orbital motion. According to Bohr's atomic model, it corresponds to the magnetic moment of an electron revolving in the first orbit (n=1) of a hydrogen atom. Its physical significance is that it serves as a natural, fundamental unit for expressing the magnetic moments of atoms, molecules, and elementary particles.
2. What is the accepted value and the SI units of the Bohr magneton?
The value of the Bohr magneton is derived from fundamental physical constants. Its accepted value and corresponding SI units are:
- Value: 9.274 × 10⁻²⁴
- SI Units: It can be expressed in two equivalent forms: Joules per Tesla (J/T) or Ampere-meter squared (A·m²). Both units are correct for measuring magnetic dipole moment.
3. How is the expression for the Bohr magneton derived for an orbiting electron?
The derivation links classical electromagnetism with Bohr's quantum condition. The steps are as follows:
- An electron with charge '-e' moving in a circle of radius 'r' at speed 'v' creates a current I = e/T, where T = 2πr/v. Thus, I = ev / 2πr.
- The magnetic moment (μ) is the product of current and area (A = πr²): μ = I × A = (ev / 2πr) × (πr²) = evr / 2.
- According to Bohr's model, the angular momentum (L) is quantised: L = mₑvr = nh / 2π, where 'n' is the principal quantum number.
- By rewriting the magnetic moment as μ = (e / 2mₑ) × (mₑvr), we get the relation μ = (e / 2mₑ) L.
- For the lowest energy state (n=1), the angular momentum is L = h / 2π. Substituting this gives the minimum magnetic moment, which is defined as the Bohr Magneton: μB = (e / 2mₑ) × (h / 2π) = eh / 4πmₑ.
4. Why is the magnetic moment of an electron due to its orbital motion considered quantised?
The magnetic moment of an orbiting electron is quantised because it is directly proportional to its orbital angular momentum, which itself is a quantised quantity. As established in the derivation, μ = (e / 2mₑ)L. Since Bohr's second postulate states that angular momentum (L) can only take discrete values (L = nh/2π), the magnetic moment (μ) must also be restricted to a set of discrete, permissible values. It cannot have any arbitrary value between these levels.
5. What is the fundamental difference between the Bohr magneton and the nuclear magneton?
The primary difference lies in the particle they describe and their resulting magnitude.
- The Bohr magneton (μB) is the unit for the magnetic moment of an electron. Its formula is μB = eh / 4πmₑ, where mₑ is the mass of the electron.
- The nuclear magneton (μN) is the unit for the magnetic moment of heavier particles like protons and neutrons. Its formula is μN = eh / 4πmₚ, where mₚ is the mass of the proton.
Since a proton is about 1836 times more massive than an electron, the nuclear magneton is approximately 1836 times smaller than the Bohr magneton.
6. Why is there a negative sign in the vector relationship between an electron's magnetic moment and its angular momentum?
The negative sign in the vector equation μ⃗ = -(e / 2mₑ) L⃗ arises due to the electron's negative charge. By convention, the direction of electric current is defined by the flow of positive charge. Since an electron is negatively charged, the direction of the equivalent current it produces is opposite to its direction of revolution. Consequently, the angular momentum vector (L⃗) and the magnetic moment vector (μ⃗) point in opposite directions.
7. In which key areas of modern physics is the concept of the Bohr magneton applied?
The Bohr magneton is a crucial concept with important applications in several fields:
- Atomic Physics: It is fundamental to explaining the Zeeman Effect, which is the splitting of atomic spectral lines when an atom is placed in an external magnetic field.
- Condensed Matter Physics: It helps in understanding and quantifying the magnetic properties of materials, such as paramagnetism, where atomic dipoles align with an external field.
- Spectroscopy: The concept is vital in techniques like Electron Paramagnetic Resonance (EPR) or Electron Spin Resonance (ESR), which detect and study unpaired electrons in substances.
