Illustrated Physics
physics, drawn
Concept

Cyclotron frequency — where it appears

The rate at which a charged particle circles in a magnetic field, qB/m, independent of its speed as long as that speed is far below light's. It sets the frequency of a cyclotron's accelerating voltage and of the radiation an electron in a field emits.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

A line carved into the X-rays of a magnetised star. A model X-ray spectrum of the kind fitted to accreting pulsars, drawn as energy per logarithmic interval on a logarithmic scale: a hard continuum that bends over above about 25 keV, with the photons near the cyclotron energy removed by resonant scattering. Dashed, the same continuum without the field. The fundamental is placed at 38 keV, where lines of this kind are seen; thermal motion of the scattering electrons along the field at kT = 8 keV broadens it to a width of about 4.8 keV. The second harmonic falls at 73.0 keV, a little below twice the fundamental. Read through ħeB/m with the relativistic level spacing and the redshift of a 1.4 solar-mass, 12 km star, z = 0.24, the fundamental says the field where the line forms is 4.24 × 10⁸ T. The depths and widths here are illustrative, not fitted to a star.

The circling that comes in quanta

An electron circling a magnetic field radiates at the frequency of its circling, and classically that is all: a continuous glow at whatever energy the circling has. On a neutron star the field is a hundred million tesla, the quantum of circling is a hard X-ray, and an electron lifted one step up falls back in a quarter of a femtosecond. So every electron there sits on the bottom step, and the X-rays that match the step are scattered out of the beam. The gap they leave is a line, and its energy is the field.

astrophysics · Radiating charge Two clocks that slow together. For a muon circling a 1.45 T magnetic field, its mean life as the laboratory measures it and the time it takes to go round once, against its Lorentz factor γ, on logarithmic scales. Time dilation stretches the life to γ × 2.197 μs: 6.6 μs at γ = 3, 64.4 μs at γ = 29.3, the muon g − 2 experiments' value. The orbit grows with the momentum, so the revolution takes γ times longer too: 15.3 ns at γ = 3, 149 ns at γ = 29.3. The two lines are parallel. Their ratio, the number of turns in one mean life, is 431 at every energy — set by the field and the muon's own properties, eBτ/2πm, with γ cancelled out.

The turns a muon makes at any energy

A muon lives two microseconds, and in a storage ring it lives longer the faster it goes, because its clock runs slow. Its orbit runs slow too: a faster muon carries more momentum, bends less in the same field and takes longer to go round. The two slowings are the same factor of gamma, so they cancel. A muon in a 1.45 tesla ring makes 431 turns in a mean life whether it is moving at nine-tenths of the speed of light or at nine nines, and the only way to more turns is a stronger magnet. The same cancellation is why the first cyclotrons stopped working near twenty million electronvolts.

relativity · Time dilation

Named alongside it

The objects these essays reach for when they reach for this one.

Decay lengthDipole fieldGravitational redshiftLandau levelsThe Lorentz factorMagnetic fieldMean lifeMuonNeutron starParticle acceleratorRadiating chargeResonant scattering

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