Question

Show that the frequency at which an electron's intrinsic magnetic dipole moment would process in a magnetic field is given by \(\omega \cong e B / m_{c}\). Calculate this frequency for a field of \(10 \mathrm{~T}\)

Short answer

The frequency at which an electron's intrinsic magnetic dipole moment would process in a 10T magnetic field is approximately \( 1.76 \times 10^{12} rad/sec \).

Step-by-step solution

Understanding the Procession Frequency

The precession frequency \( \omega \) of an electron in a magnetic field is given by the formula \( \omega = \gamma B \), where \( \gamma \) is the gyromagnetic ratio, and \( B \) is the magnetic field. The gyromagnetic ratio for an electron is given by \( \gamma = \frac{e}{m_{c}} \), where \( e \) is the elementary charge and \( m_{c} \) is the mass of the electron. Substituting this into the previous formula gives us \( \omega = \frac{e}{m_{c}} B \).

Simplifying the Expression

The expression can be simplified as \( \omega \cong \frac{e B}{m_{c}}\). This is the frequency at which an electron's intrinsic magnetic dipole moment would process in a magnetic field.

Calculating the Precession Frequency

Given that \( B = 10 T \) (Tesla), the frequency can be calculated by substituting \( B \) into the equation: \( \omega = \frac{e \cdot 10}{m_{c}} \). The elementary charge, \( e \), is approximately \( 1.602 \times 10^{-19} C \) (Coulombs), and the electron mass, \( m_{c} \), is approximately \( 9.109 \times 10^{-31} Kg \). Substituting these values gives: \( \omega = \frac{1.602 \times 10^{-19} \cdot 10}{9.109 \times 10^{-31}} = 1.76 \times 10^{12} rad/sec \). This is the precession frequency of the electron's magnetic dipole moment in the given field.

Key concepts

Gyromagnetic Ratio

The gyromagnetic ratio is a fundamental property that relates the magnetic moment and angular momentum of a particle. For electrons, this is a key element when discussing their behavior in a magnetic field.

In physics, the gyromagnetic ratio, often denoted as \( \gamma \), is the ratio of the magnetic dipole moment to the angular momentum. Specifically for an electron, the expression is given by \( \gamma = \frac{e}{m_c} \), where:

• \( e \) is the elementary charge \( (1.602 \times 10^{-19} \text{ C}) \)
• \( m_c \) is the mass of the electron \( (9.109 \times 10^{-31} \text{ kg}) \)

This ratio is crucial in determining how electrons behave under the influence of a magnetic field as it directly links the magnetic dipole moment and the precession frequency, giving rise to phenomena like Electron Paramagnetic Resonance (EPR).

Understanding \( \gamma \) helps provide insight into the fundamental interactions between magnetic fields and quantum mechanical electrons.

Magnetic Dipole Moment

The magnetic dipole moment is an essential concept in understanding electron behavior in magnetic fields. A dipole moment occurs whenever there's a separation of positive and negative charges within a system.

For electrons, the intrinsic magnetic dipole moment arises from their spin and charge, contributing to their overall magnetic properties. This moment is analogous to tiny magnets within electrons and is represented mathematically as a vector.

• It contributes to an electron's response in magnetic fields.
• This moment affects how electrons align themselves in magnetic fields, leading to precession, a rotational movement around the magnetic field lines.

The precession frequency can be calculated by relating the magnetic dipole moment with the applied magnetic field, showcasing the direct connection in quantum physics.

This core concept illustrates how electrons interact at a microscopic level, which can drastically impact material properties on a larger scale.

Electron in Magnetic Field

An electron placed in a magnetic field exhibits fascinating behavior due to its intrinsic properties such as charge and magnetic dipole moment.

When subjected to a magnetic field, an electron's magnetic dipole attempts to align with the field. However, due to its quantum nature, this alignment results in precession. This is similar to how a spinning top wobbles around its vertical axis.

• The precession of electrons is a critical concept in fields such as nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI).
• Its frequency is determined by factors such as the strength of the magnetic field and the gyromagnetic ratio of the electron.

In the exercise, it was shown that an electron in a \( 10 \) Tesla magnetic field has a precession frequency of approximately \(1.76 \times 10^{12} \text{ rad/sec}\).

Understanding this behavior helps bridge the gap between quantum mechanics and practical application in technology and materials science.