Charged particles with intrinsic spin or orbital angular momentum act like tiny magnets. When placed in an external magnetic field, they experience a torque that causes the magnetic moment to precess around the field direction. This motion, known as Larmor precession, plays a central role in technologies like nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI), and electron spin resonance. The precession frequency depends on the particle’s charge-to-mass ratio and the strength of the magnetic field.
For a particle of charge and mass in a uniform magnetic field , the angular frequency of precession is given by . Multiplying by converts it to cycles per second, or hertz. This simple relation ignores subtle quantum effects like anomalous magnetic moments, but it captures the essential physics for many applications.
The form is pre-populated with constants for the electron, one of the most common particles studied in magnetic resonance. You may substitute values for protons, neutrons, or any charged species. After entering a magnetic field magnitude, click Compute to display the angular frequency in radians per second and the ordinary frequency in hertz. Because the formula involves only multiplication and division, the computation is nearly instantaneous.
NMR and MRI rely on the fact that atomic nuclei precess at characteristic frequencies in magnetic fields. By applying carefully tuned radio-frequency pulses, scientists and doctors can manipulate these nuclear spins to probe the structure of molecules or produce detailed images of the human body. The Larmor frequency determines which radio frequency will resonate with a given nucleus. For example, a proton in a 1 T field precesses at about 42.6 MHz. Adjusting the field strength or selecting different nuclei leads to different resonance frequencies.
In classical physics, you can picture the magnetic moment vector sweeping out a cone around the magnetic field line. Quantum mechanics introduces discrete energy levels associated with different spin orientations, but the expectation value of the spin still precesses according to the classical formula. This duality between classical motion and quantum energy splitting underpins magnetic resonance phenomena. When a resonance condition is met, the system can absorb or emit photons at the Larmor frequency.
The precession rate scales with the absolute value of the charge divided by twice the mass. Lighter particles with larger charges precess more rapidly. This sensitivity enables precise measurements of charge-to-mass ratios in physics experiments. Deviations from the expected frequency can also reveal tiny shifts in a particle’s magnetic moment due to interactions with the environment or new physics beyond the standard model.
Joseph Larmor formulated the concept of precession in 1897 while exploring the relationship between electron orbits and magnetic fields. His theoretical insight paved the way for modern magnetic resonance techniques. In the 1940s, physicists discovered how to exploit nuclear precession to measure magnetic fields with remarkable accuracy. These discoveries eventually led to the development of MRI scanners, which revolutionized medical diagnostics by providing non-invasive images of soft tissue.
Larmor precession is not limited to medicine. It influences the behavior of plasmas in fusion experiments, the dynamics of particles trapped in magnetic bottles, and the design of precision magnetometers. Scientists studying fundamental symmetries of nature, such as the search for the neutron electric dipole moment, rely on precise measurements of precession frequencies. Because the motion is so sensitive to the surrounding magnetic field, Larmor precession provides a window into both basic physics and practical technology.
After running the calculator, you will see both angular frequency and ordinary frequency. If you increase the magnetic field, the frequency rises linearly, doubling when the field doubles. Similarly, changing to a lighter particle or one with greater charge increases the precession rate. These trends explain why electrons precess much faster than protons in the same field. When planning an experiment or designing an NMR protocol, use these calculations to anticipate the resonance frequencies you will encounter.
The simple formula used here assumes the particle’s magnetic moment is exactly . In reality, quantum corrections lead to slight deviations characterized by the g-factor. Electrons have a g-factor near 2.0023, while protons are about 5.585. For high-precision work, replace with . The calculator focuses on the classical approximation, making it ideal for introductory learning and quick estimates.
Larmor precession illuminates how quantum systems interact with magnetic fields. By experimenting with different parameters in the calculator, you can deepen your understanding of this fundamental phenomenon. Whether you are studying magnetic resonance imaging, particle physics, or condensed matter systems, mastering the Larmor frequency is key to predicting the behavior of spinning charges in magnetic environments.
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