Frank-Tamm formula

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The Frank-Tamm formula yields the amount of Cherenkov radiation emitted as a charge particle moves through a medium at superluminal velocity.

The Energy emitted per unit length per unit frequency is:

$dE = \frac{\mu q^2} {4 \pi} \omega (1 - \frac{c^2} {v^2 n(\omega)^2}) dx d\omega$ ,

where $μ$ and $n (ω)$ are the permeability and index of refraction of the medium, $q$ is the charge of the particle, $v$ is the speed of the particle, $c$ is the speed of light in vacuum, and $ω$ is the angular frequency of radiation.

The total amounted of energy radiated per unit length is:

$\frac{dE}{dx} = \int_{v>c/n(\omega)} \frac{\mu q^2} {4 \pi} \omega (1 - \frac{c^2} {v^2 n(\omega)^2}) d\omega$