Larmor Precession Reference Cheat Sheet

Key Facts

• The magnetic torque on a magnetic dipole is $\vec{\tau} = \vec{\mu} \times \vec{B}$, so its magnitude is $\tau = \mu B\sin\theta$.
• The magnetic potential energy of a dipole in a uniform magnetic field is $U = -\vec{\mu}\cdot\vec{B} = -\mu B\cos\theta$.
• If the magnetic moment is proportional to angular momentum, the relation is $\vec{\mu} = \gamma \vec{L}$, where $\gamma$ is the gyromagnetic ratio.
• The Larmor angular frequency has magnitude $\omega_L = |\gamma|B$, where $B$ is the magnetic field strength.
• The Larmor frequency in cycles per second is $f_L = \frac{\omega_L}{2\pi} = \frac{|\gamma|B}{2\pi}$.
• For a particle with charge $q$, mass $m$, and $g$-factor $g$, the gyromagnetic ratio is $\gamma = \frac{gq}{2m}$.
• The precession direction depends on the sign of $\gamma$, so positive and negative charges precess in opposite senses around $\vec{B}$.
• Magnetic resonance occurs when an applied wave satisfies $hf = \Delta E$, often matching $f = f_L$ for spin transitions.

Vocabulary

Larmor precession

Larmor precession is the steady rotation of a magnetic moment or angular momentum vector around an external magnetic field.

Magnetic moment

A magnetic moment $\vec{\mu}$ measures how strongly an object behaves like a tiny magnet and how it interacts with $\vec{B}$.

Gyromagnetic ratio

The gyromagnetic ratio $\gamma$ is the constant relating magnetic moment to angular momentum through $\vec{\mu} = \gamma \vec{L}$.

Angular frequency

Angular frequency $\omega$ measures rotational rate in radians per second, with $\omega = 2\pi f$.

Torque

Torque $\vec{\tau}$ is a twisting effect that changes angular momentum according to $\vec{\tau} = \frac{d\vec{L}}{dt}$.

Resonance

Resonance occurs when an applied oscillating field has the correct frequency to transfer energy efficiently, such as $hf = \Delta E$.