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Azurwake

Cherenkov Radiation / Light-Speed in Water Physics

\[\cos\theta = \frac{1}{n\beta}, \qquad \beta > \frac{1}{n}\]
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Azurwake is a Pixelated Physics law character, one of our forces designs. Cherenkov radiation: a charged particle outrunning light in water emits a cone of blue light, like an optical sonic boom. This is a 3" × 3" pixel-art sticker. The physics panel below gives the equation, what each symbol means and the sources.

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Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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The physics

A charged particle moving through a medium faster than light travels in that medium emits light in a cone, like a sonic boom. The cone angle depends on the particle's speed and the refractive index, and Frank and Tamm's formula gives how many photons come out per unit length.

Azurwake is breaking the speed limit, and it is perfectly legal.

Nothing outruns light in a vacuum. But in water, light travels at c/n, about three-quarters of c, because water’s refractive index is about 1.33. A fast enough charged particle can beat that. An electron needs a speed of only about 0.75c, roughly a quarter of an MeV of kinetic energy. Then it is faster than light travels in water, and the water answers with a cone of blue light.

The geometry is that of a jet’s sonic boom, which forms only when the jet outruns sound. Each point on the particle’s track sends out a wavelet of light; when the source outruns its own waves, the wavelets pile up along a cone. The light leaves at a fixed angle to the path, the first line on this page: for a particle near light speed in water, about 41°. The shock front itself, the V whose apex rides on the electron at the bow, has a half-angle of 90° minus that, about 49°.

Pavel Cherenkov, working under Sergei Vavilov at the Lebedev Institute of Physics, saw the faint blue glow in 1934 while studying pure liquids under gamma rays. In 1937 Ilya Frank and Igor Tamm explained it with classical electrodynamics. Their formula, the second line, gives the number of photons per length and per wavelength. It grows towards short wavelengths, which is why the glow looks blue-violet. The three shared the 1958 Nobel Prize.

It is the bluish glow in the water pools that shield some nuclear reactors. She never exceeds c, only the light in the water.

Faster than light. In water. Legally.

Equations

\[\cos\theta = \frac{1}{n\beta}, \qquad \beta > \frac{1}{n}\]
\[\begin{aligned} \frac{d^2N}{dx\,d\lambda} &= \frac{2\pi\alpha z^2}{\lambda^2} \\ &\quad\times\left(1-\frac{1}{\beta^2 n^2}\right) \end{aligned}\]
SymbolMeaningUnit
\(\theta\) angle between the emitted light and the particle's path (the wavefront cone has half-angle 90° − θ) \(\mathrm{rad}\)
\(n\) refractive index of the medium, about 1.33 for water \(\mathrm{1}\)
\(\beta\) particle speed as a fraction of c \(\mathrm{1}\)
\(N\) number of photons emitted \(\mathrm{1}\)
\(x\) distance travelled by the particle \(\mathrm{m}\)
\(\lambda\) wavelength of the emitted light \(\mathrm{m}\)
\(\alpha\) fine-structure constant, about 1/137 \(\mathrm{1}\)
\(z\) charge of the particle in units of e \(\mathrm{1}\)

Sources

  1. Cherenkov (1934) Visible emission of clean liquids by action of γ radiation, Dokl. Akad. Nauk SSSR 2, 451 — reprinted in Usp. Fiz. Nauk 93, 385 (1967)
  2. Frank & Tamm (1937) Coherent visible radiation of fast electrons passing through matter, Dokl. Akad. Nauk SSSR 14, 109 — reprinted in Usp. Fiz. Nauk 93, 388 (1967)
  3. NobelPrize.org — The Nobel Prize in Physics 1958 (Cherenkov, Frank and Tamm)
  4. Encyclopaedia Britannica — Cherenkov radiation
  5. NobelPrize.org — Pavel A. Cherenkov, Biographical (1934 observation while working under S. I. Vavilov; Lebedev Institute of Physics)

Source Frank, I.; Tamm, I. (1937). Coherent visible radiation of fast electrons passing through matter. Doklady Akademii Nauk SSSR 14, 109. Reprinted in Usp. Fiz. Nauk 93, 388–393 (1967),doi.org/10.3367/UFNr.0093.196710o.0388

Field notes

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