Problem 13 - Classical Mechanics

Question

The angular velocity of the electron in the lowest Bohr orbit of the hydrogen atom is approximately \(4 \times 10^{16} \mathrm{~s}^{-1}\). What is the largest magnetic field which may be regarded as small in this case, in the sense of \(\S 5.5 ?\) Determine the Larmor frequency in a field of \(2 \mathrm{~T}\).

Step-by-Step Solution

Answer

In this problem, we were asked to find the largest magnetic field that can be considered small for an electron in the lowest Bohr orbit of a hydrogen atom and calculate the Larmor frequency in a field of \(2 \mathrm{~T}\). We calculated the largest magnetic field that can be considered small as \(1.96 \times 10^{-4} \,\mathrm{T}\) and the Larmor frequency in a \(2\,\mathrm{T}\) magnetic field as \(2.8 \times 10^{11}\,\mathrm{s^{-1}}\).

Step 1: Calculate the electron's magnetic moment.

We have to find the magnetic moment of the electron first. The magnetic moment of an electron is given by the formula: \(\boldsymbol{\mu} = -\dfrac{e}{2m_{e}}\boldsymbol{L}\) where \(e\) is the charge of the electron, \(m_{e}\) is the mass of the electron, and \(\boldsymbol{L}\) is the angular momentum of the electron. The angular momentum of the electron is related to its angular velocity by the equation: \(\boldsymbol{L} = m_{e}r^2\boldsymbol{\omega}\) where \(r\) is the radius of the lowest Bohr orbit and \(\boldsymbol{\omega}\) is the angular velocity. Given the angular velocity of the electron, we can calculate the magnetic moment as follows: \(\boldsymbol{\mu} = -\dfrac{e}{2m_{e}}(m_{e}r^2\boldsymbol{\omega})\)

Step 2: Determine the largest magnetic field that can be considered small.

Now we need to find the largest magnetic field under which the electron's magnetic moment will still precess slowly (in the sense of section 5.5). In order for the magnetic field to be considered small, the Larmor frequency should be much smaller than the electron's angular frequency, which corresponds to the condition \(\omega_{L} \ll \boldsymbol{\omega}\) where \(\omega_{L}\) is the Larmor frequency. The Larmor frequency is given by the formula: \(\omega_{L} = \dfrac{|\boldsymbol{\mu}|B}{\hbar}\) Substituting the magnetic moment expression, we get: \(\omega_{L} = \dfrac{-e}{2m_e}(m_{e}r^2\boldsymbol{\omega})B \div \hbar\) For \(\omega_{L} \ll \boldsymbol{\omega}\), we can write the condition: \(B \ll \dfrac{2\hbar\boldsymbol{\omega}}{-e(r^2)}\) Now we have all the quantities we need to find the largest magnetic field that can be considered small. Substituting the given angular velocity and the values of the constants \(e\) (elementary charge), \(\hbar\) (reduced Planck constant), and \(r\) (Bohr radius), we can calculate the largest magnetic field: \(B \ll \dfrac{2(1.054 \times 10^{-34}\,\mathrm{J\cdot s})(4 \times 10^{16}\,\mathrm{s}^{-1})}{1.602 \times 10^{-19}\,\mathrm{C}(5.29 \times 10^{-11}\,\mathrm{m})^2}\) \(B \ll 1.96 \times 10^{-4} \,\mathrm{T}\) So, the largest magnetic field that may be regarded as small is \(1.96 \times 10^{-4} \,\mathrm{T}\).

Step 3: Determine the Larmor frequency in a \(2\,\mathrm{T}\) magnetic field.

Now we need to find the Larmor frequency of the electron in a magnetic field of strength \(2\,\mathrm{T}\). We can use the Larmor frequency equation derived in step 2: \(\omega_{L} = \dfrac{-e}{2m_e}(m_{e}r^2\boldsymbol{\omega})B \div \hbar\) Substituting the given magnetic field strength and the values of the constants \(e\), \(\hbar\), \(\boldsymbol{\omega}\), and \(r\), we can calculate the Larmor frequency: \(\omega_{L} = \dfrac{-1.602 \times 10^{-19}\,\mathrm{C}}{2(9.109 \times 10^{-31}\,\mathrm{kg})}((9.109 \times 10^{-31}\,\mathrm{kg})(5.29 \times 10^{-11}\,\mathrm{m})^2(4 \times 10^{16}\,\mathrm{s}^{-1}))(2\,\mathrm{T}) \div (1.054 \times 10^{-34}\,\mathrm{J\cdot s})\) \(\omega_{L} = 2.8 \times 10^{11}\,\mathrm{s^{-1}}\) So, the Larmor frequency of the electron in a \(2\,\mathrm{T}\) magnetic field is \(2.8 \times 10^{11}\,\mathrm{s^{-1}}\).

Key Concepts

Bohr Model

The Bohr model is a classical model of the hydrogen atom proposed by Niels Bohr in 1913. It provides an explanation for the stability of atoms and the discrete levels of energy that electrons can occupy. According to this model, electrons move in fixed circular orbits around the nucleus, with each orbit corresponding to a specific energy level.

The Bohr model takes Planck's quantum theory into account, allowing only certain values of angular momentum, which is quantized in units of the reduced Planck's constant, \(\hbar\). The radius of these orbits and the energy associated with them can be determined using the quantum condition that the angular momentum is an integer multiple of \(\hbar\).

In calculations involving the Bohr model, the electrostatic force between the electron and the nucleus is balanced by the centripetal force required for electron motion in a circular path, leading to an expression for the radius of the orbit, the electron's velocity, and hence its angular velocity. This model, although not fully accurate when compared with modern quantum mechanics, provides essential insights into the quantized nature of atomic systems and is crucial for understanding phenomena such as spectral lines.

Magnetic Moment

The magnetic moment of a particle is a fundamental property related to its magnetic field and the torque it would experience in the presence of an external magnetic field. For an electron in an atom, the magnetic moment arises from two sources: its orbital motion around the nucleus and its intrinsic spin.

The formula for the magnetic moment due to the electron's orbital motion is given by \(\mu = -\frac{e}{2m_e} L\), where \(e\) is the magnitude of the electron's charge, \(m_e\) is the electron's mass, and \(L\) is its angular momentum. Since the angular momentum is quantized in the Bohr model, the magnetic moment is also inherently quantized.

When an external magnetic field is applied, the magnetic moment interacts with it, leading to a phenomenon called Larmor precession, wherein the direction of the magnetic moment precesses around the magnetic field. The strength of this interaction is characterized by the Larmor frequency, providing a direct connection between the magnetic moment and observable effects like resonance frequencies in magnetic fields.

Angular Velocity

Angular velocity is a vector quantity that represents the rate of change of the angle of rotation of an object. For an electron in an atom, the concept specifically refers to how quickly the electron travels around the nucleus in its orbit.

In the context of the Bohr model, the angular velocity of an electron in its orbit is determined by the orbital radius and the energy level it is in. Mathematically, it is represented by the symbol \(\omega\) and is related to the angular momentum of the electron by the equation \(L = m_e r^2 \omega\).

The magnitude of the angular velocity is instrumental in calculations pertaining to the magnetic moment and the Larmor frequency. It serves as a fundamental link connecting the motion of the electron to the magnetic properties of the atom. Understanding the relationship between angular velocity and other atomic properties allows us to calculate behaviors like the precession rate of the magnetic moment in the presence of external magnetic fields, which is vital in applications such as magnetic resonance imaging (MRI) and other spectroscopic techniques.