Larmor-Coriolis analogy

The rotating frame at the Larmor frequency is arguably the most-used non-inertial frame in all of technology. Every MRI scanner begins its pulse-sequence mathematics by transforming into the frame precessing with the proton spins ($42.58\ \mathrm{MHz/T}$ for protons — the orbital $q B/2m_p$ dressed with the proton's g-factor of 5.59); in that frame the enormous static field vanishes from the equations, exactly as Larmor's theorem promises, and the faint radio-frequency pulses that flip spins become simple static fields an experimenter can reason about. The entire craft of NMR spectroscopy — spin echoes, decoupling, two-dimensional sequences — is choreography written in Larmor's rotating frame. The analogy also runs forward, not just backward. Optically pumped magnetometers — the sensors now replacing liquid-helium SQUIDs in wearable magnetoencephalography helmets — measure magnetic fields by tracking the Larmor precession of alkali atoms optically, reaching femtotesla sensitivity at room temperature: the 2026-era brain scanner is literally a Larmor-frequency counter strapped to your head. And cold-atom laboratories run the dictionary in the direction Larmor never imagined: rotate a neutral Bose–Einstein condensate and the Coriolis force on the neutral atoms is a synthetic magnetic field, complete with quantized vortices playing the role of flux lines — engineered magnetism for particles with no charge at all, the analogy of this article promoted from mnemonic to manufacturing technique.

Roll a ball across a spinning carousel and it curves sideways, pushed by no one — the Coriolis force, a pure artifact of your rotating coordinates. Fire an electron across a magnetic field and it curves sideways too, pushed by the very real magnetic force. Put the two trajectories side by side and you cannot tell which is which: both deflections are proportional to the speed, both are at right angles to the motion, and neither ever changes the particle's kinetic energy. The Coriolis force article derives the first; electromagnetism supplies the second; this article is about what you can do with the fact that they are the same shape.

The power move is to run the analogy in reverse. On a carousel, the sideways force appears because you insisted on rotating coordinates; step off the platform and it is gone. So if a magnetic force has the same shape as a Coriolis force, you should be able to make it disappear by deliberately climbing onto the right turntable. There is exactly one rotation rate that works. The Coriolis coefficient is $2m\omega$ while the magnetic coefficient is $qB$, so the cancelling frame must rotate at $\omega = qB/2m$ — the Larmor frequency. In that co-rotating frame a bound electron feels no magnetic force at all (to first order), which is why an atom placed in a magnet does not squash or stretch: its electron cloud simply precesses, rigidly, like a tilted gyroscope, at $\omega_L$.

Why only to first order? Because a rotating frame charges rent. Climbing onto the turntable cancels the Coriolis-shaped magnetic force, but it also introduces a centrifugal force $\propto \omega_L^2$, and the magnetic field brings no matching $B^2$ force of its own to pay that bill. The mismatch — small as $B^2$ — is real physics, not bookkeeping failure: it is the classical origin of diamagnetism, the weak universal magnetism by which every atom slightly opposes an applied field. The analogy even predicts its sign: the leftover term acts like a tether pulling orbits toward the field axis, and by Lenz-law logic the induced circulation opposes $\mathbf{B}$. Keep one contrast in view throughout: for a bound charge (an electron in an atom), the binding force is enormous and the $B^2$ residue is a negligible perturbation — Larmor's theorem is then essentially exact. For a free charge there is no binding force at all, the "small" residual becomes the entire centripetal budget, and the particle orbits at the cyclotron frequency $\omega_c = qB/m$ — exactly twice the Larmor rate. The factor of two that puzzles every student is not a convention clash; it is the difference between cancelling a force and being steered by it.

Start from the two force laws whose resemblance drives everything. A particle observed in a frame rotating at constant angular velocity $\boldsymbol{\omega}$ obeys Newton's law with fictitious forces appended — this is equation (3) of the Coriolis force article, quoted here as the mechanical side of the dictionary:

where: $\mathbf{F}$ — the sum of real forces; the three correction terms are the Coriolis, centrifugal, and Euler forces, in that order. Each is pure kinematics — the price of using turning coordinates. The Coriolis term is the one with the magnetic personality: linear in $\mathbf{v}_{\mathrm{rot}}$ and perpendicular to it. On the electromagnetic side sits the Lorentz force — a genuine interaction, measured in any frame:

where: $q$ — the particle's charge; $\mathbf{E}, \mathbf{B}$ — the electric and magnetic fields. The magnetic piece $q\,\mathbf{v}\times\mathbf{B}$ shares every structural property of the Coriolis force: it vanishes for a particle at rest, it is perpendicular to the velocity, and it therefore does no work — a magnetic field can bend a trajectory but never change a speed. Demand now that a Coriolis force impersonate the magnetic force for every possible velocity: $-2m\,\boldsymbol{\omega}\times\mathbf{v} = q\,\mathbf{v}\times\mathbf{B} = -q\,\mathbf{B}\times\mathbf{v}$. Since this must hold as an identity in $\mathbf{v}$, the coefficients of $(\cdot)\times\mathbf{v}$ must match: $2m\,\boldsymbol{\omega} = q\,\mathbf{B}$. The rotation rate that lets a frame generate the magnetic force is $+q\mathbf{B}/2m$; the frame that cancels it rotates oppositely, and that cancelling rate is the one honored with the name:

where: $\boldsymbol{\omega}_L$ — the Larmor frequency, a vector along the field axis. For a positive charge the precession is clockwise seen from the tip of $\mathbf{B}$; for an electron ($q = -e$) it is counterclockwise, with magnitude $\omega_L = eB/2m_e$. Notice what is absent from (3): the binding force, the orbit radius, the orbit shape. One field, one charge-to-mass ratio, one universal precession rate — that universality is why the Zeeman effect splits spectral lines cleanly rather than smearing them. The identification extends line by line, and it is worth having the whole dictionary on one card:

The residual term in (4) is worth a second look before moving on, because it is the entire classical theory of diamagnetism in one line. It is quadratic in $B$, always axis-seeking, and cannot be transformed away by any frame choice. The precession it accompanies turns each electron orbit into a slow circulating current whose magnetic moment opposes the applied field — Langevin's 1905 result, $\Delta\boldsymbol{\mu} = -\frac{q^2\langle r_\perp^2\rangle}{4m}\,\mathbf{B}$, negative and temperature-independent, exactly as measured for filled-shell atoms. Where the centrifugal force flings carousel riders outward, its electromagnetic twin quietly makes every substance in the universe very slightly magnet-averse.

Which formula answers which question

Larmor's theorem makes a spectroscopic prediction with no adjustable parameters. Model an atomic electron classically as an isotropic oscillator of frequency $\omega_0$ — Lorentz's model, state of the art in 1897. Decompose its motion into one linear oscillation along $\mathbf{B}$ and two counter-rotating circular motions in the perpendicular plane. The parallel component involves velocities along $\mathbf{B}$, feels no magnetic force, and radiates at $\omega_0$ unchanged. The two circular components are orbits in the plane where Larmor's theorem operates: seen from the lab, one is dragged with the precession and one against it, shifting their frequencies to $\omega_0 + \omega_L$ and $\omega_0 - \omega_L$:

where: $\Delta\omega$ — the splitting between adjacent lines of the classical Zeeman triplet. The prediction is stark: every spectral line, of every element, should split in a magnetic field into three components separated by the same universal $\omega_L$ — because equation (3) knows only $e/m$, not the atom. For many lines (those between singlet states, it turned out) this is exactly what is observed, polarizations and all; the "anomalous" patterns that disobey it were the first experimental fingerprints of electron spin, a genuinely quantum ingredient the classical analogy cannot supply. Historical bonus: from the measured splitting Lorentz and Zeeman extracted $e/m_e$ in 1896 — a year before Thomson's cathode-ray measurement, and it matched.

1 · The electron's two frequencies in a 1.0-tesla field (confidence builder)

For an electron in a uniform field $B = 1.0\ \mathrm{T}$, find the Larmor frequency and the cyclotron frequency in Hz, and check whether Larmor's theorem is valid for an electron bound in a hydrogen atom at this field.

Step 1. Larmor frequency from equation (3): $\omega_L = \dfrac{eB}{2m_e} = \dfrac{(1.602\times10^{-19})(1.0)}{2\,(9.109\times10^{-31})} = 8.79\times10^{10}\ \mathrm{rad/s}$, so $f_L = \omega_L/2\pi = \boxed{14.0\ \mathrm{GHz}}$.

Step 2. Cyclotron frequency from equation (6): $\omega_c = 2\,\omega_L = 1.76\times10^{11}\ \mathrm{rad/s}$, so $f_c = \boxed{28.0\ \mathrm{GHz}}$ — the number microwave engineers know as the electron cyclotron resonance at 1 T.

Step 3. Validity check via equation (5). The orbital frequency of a ground-state hydrogen electron is $\omega_0 = v/r = (2.19\times10^{6})/(5.29\times10^{-11}) = 4.1\times10^{16}\ \mathrm{rad/s}$. The ratio is $\omega_L/\omega_0 = 8.79\times10^{10}/4.1\times10^{16} = 2.1\times10^{-6}$, so the neglected residual is $(\omega_L/\omega_0)^2 = \boxed{4.5\times10^{-12}}$ of the binding force.

Step 4. Interpret: at one tesla — already a strong laboratory magnet — the atom's internal dynamics outpaces the Larmor precession by a factor of half a million, and the "approximation" in Larmor's theorem is one part in a trillion. The atom precesses as rigidly as any theorem could ask.

2 · The classical Zeeman triplet of the cadmium red line (exam level)

The cadmium line at $\lambda = 643.8\ \mathrm{nm}$ — the line on which Zeeman's effect was first cleanly resolved — is observed in a field $B = 1.5\ \mathrm{T}$. Find the splitting of the classical triplet in frequency and in wavelength, and the spectral resolving power needed to see it.

Step 1. The classical triplet (equation (8)) is split by the Larmor frequency. At $1.5\ \mathrm{T}$: $\Delta\nu = f_L = 14.0\ \mathrm{GHz/T} \times 1.5\ \mathrm{T} = \boxed{21.0\ \mathrm{GHz}}$.

Step 2. Convert to wavelength. Differentiating $\nu = c/\lambda$ gives $|\Delta\lambda| = \lambda^2\,\Delta\nu/c = \dfrac{(643.8\times10^{-9})^2\,(2.10\times10^{10})}{3.00\times10^{8}}$. With $\lambda^2 = 4.14\times10^{-13}\ \mathrm{m^2}$: $\Delta\lambda = 4.14\times10^{-13} \times 2.10\times10^{10} / 3.00\times10^{8} = \boxed{0.029\ \mathrm{nm}}$ — about 29 picometres between adjacent components.

Step 3. Resolving power: $R = \lambda/\Delta\lambda = 643.8/0.029 \approx \boxed{22{,}000}$. A good grating spectrograph or a Fabry–Pérot etalon reaches this comfortably; Zeeman's 1896 grating (Rowland, $R \sim 10^5$) had margin to spare. A pocket spectroscope ($R \sim 10^3$) would see only a slight fattening of the line — which is exactly how the effect first announced itself.

Step 4. The kicker: the splitting contains $e/m_e$ and nothing else about cadmium. Inverting Step 1, a measured $\Delta\nu/B$ of $14.0\ \mathrm{GHz/T}$ gives $e/m_e = 4\pi\,\Delta\nu/B\cdot\tfrac{1}{2}\cdot\ldots$ — more transparently, $e/m_e = 2\omega_L/B = 1.76\times10^{11}\ \mathrm{C/kg}$. Lorentz and Zeeman ran precisely this inversion in 1896 and beat Thomson's cathode-ray value of $e/m$ to press by a year.

References. Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., §5.9 (Larmor's theorem) · Landau & Lifshitz, Mechanics, 3rd ed., §39 problems · Morin, Introduction to Classical Mechanics, Ch. 10 · Taylor, Classical Mechanics, Ch. 9 · Jackson, Classical Electrodynamics, 3rd ed., §5.7 (diamagnetism, Larmor frequency) The key results thread together as follows: the coefficient $2m$ in the Coriolis force gives $\omega_L=qB/2m=\tfrac12\omega_c$; Derivation 1 yields the single residual $O(B^2)$ term; Derivation 2 gives the cyclotron solution $\omega_c=qB/m$; Derivation 3 gives $\dot{\boldsymbol{\mu}}=\boldsymbol{\omega}_L\times\boldsymbol{\mu}$ with $\gamma=q/2m$. The worked-example numbers ($f_L = 14.0\ \mathrm{GHz}$ and $f_c = 28.0\ \mathrm{GHz}$ at $1\ \mathrm{T}$; $\Delta\lambda = 0.029\ \mathrm{nm}$ for Cd 643.8 nm at 1.5 T; $(\omega_L/\omega_0)^2 = 4.5\times10^{-12}$ for hydrogen at 1 T) follow directly from the boxed formulas.

Builds on: Coriolis force · Centrifugal force · Time derivatives in rotating frames · Angular velocity vector of a rotating frame · Fictitious (inertial) forces. Leads to: Effective gravity on the rotating Earth · Euler force.