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Bremsella

Bremsstrahlung (Braking Radiation) Physics

\[P = \frac{q^2 a^2}{6\pi\varepsilon_0 c^3}\]
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Where do X-rays come from? Bremsella is an electron swerving past a nucleus. Any accelerating charge radiates, and for slow charges the Larmor formula gives the power, P = q²a²/(6πε₀c³), so doubling the acceleration quadruples it. Bremsstrahlung, German for braking radiation, gives an X-ray tube its continuous spectrum as fast electrons stop in the metal target. A 3" pixel-art sticker; equation, symbols and sources in the physics panel below.

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The physics

Any accelerating charge radiates. Larmor's formula says the power grows with the square of the charge and the square of its acceleration, which is why electrons that slow down sharply in matter give off braking radiation.

Bremsella is an electron that takes the corner past a nucleus too fast and slams on the brakes.

Any accelerating charge radiates. For speeds well below c the power is given by the Larmor formula, P = q²a²/(6πε₀c³). Double the deceleration and the power goes up four times.

On the sticker the dashed trail bends around the pink nucleus and amber photons fly off the braking electron. Bremsstrahlung is German for braking radiation.

It is how an X-ray tube makes its continuous spectrum: fast electrons stopping in a metal target. Larmor gives only the total power, and tube electrons reach about 0.4–0.6c, so the spectrum's shape needs the full theory (Kramers, Bethe–Heitler), with a sharp cutoff at hν_max = eV, the Duane–Hunt limit.

Slow down. Light up.

Equations

\[P = \frac{q^2 a^2}{6\pi\varepsilon_0 c^3}\]
SymbolMeaningUnit
\(P\) power radiated by the accelerating charge (non-relativistic Larmor formula) \(\mathrm{W}\)
\(q\) charge of the particle \(\mathrm{C}\)
\(a\) magnitude of its acceleration \(\mathrm{m/s^2}\)
\(\varepsilon_0\) vacuum permittivity \(\mathrm{F/m}\)
\(c\) speed of light \(\mathrm{m/s}\)

Sources

  1. Larmor, J. (1897) On a dynamical theory of the electric and luminiferous medium, Phil. Mag. 44, 503
  2. Jackson, J. D., Classical Electrodynamics, 3rd ed., Ch. 14 (Larmor formula) and Ch. 15 (bremsstrahlung)
  3. Wikipedia, Bremsstrahlung

Source Larmor, J. (1897). On the theory of the magnetic influence on spectra; and on the radiation from moving ions. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 44, 503–512.doi.org/10.1080/14786449708621095

Field notes

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