The energy d E emitted per unit length travelled by the particle per unit of frequency d ω ω is: ∂ ∂ 2 E ∂ ∂ x ∂ ∂ ω ω = q 2 4 π π μ μ ( ω ω ) ω ω ( 1 − − c 2 v 2 n 2 ( ω ω ) ) provided that β β = v c > 1 n ( ω ω ) . Here μ μ ( ω ω ) and n ( ω ω ) are the frequency-dependent permeability and index of refraction of the medium respectively, q is the electric charge of the particle, v is the speed of the particle, and c is the speed of light in vacuum.