Field of a moving particle in a dielectric The E and B field associated to a uniformly moving particle in vacuum are given by and where Previously we considered the case b 1 P Piot PHYS 571 Fall 2007 Field in the limit b 1 We now consider the extreme case b 1 The modified Bessel functions have the asymptotic expansion And the field associated to a uniformly moving charged particle takes the form P Piot PHYS 571 Fall 2007 Cerenkov condition We saw that to get radiation or energy loss we needed either or to be complex When investigating dielectric screening effects we considered the case C We now consider the case where is a pure imaginary number and R from P Piot PHYS 571 Fall 2007 Cerenkov condition 0 Consider the model for permittivity 1 2 1 L 0 P Piot PHYS 571 Fall 2007 Energy loss Explicit the asymptotic form of the e m field in the energy loss equation gives P Piot PHYS 571 Fall 2007 Frank Tamm energy loss formula But I so 1 so finally This is Frank Tamm formula derived in 1937 History Cerenkov observed the radiation in Vavilov s labs 1934 Frank and Tamm explained the effect 1937 Cerenkov Frank and Tamm share Nobel prize 1958 P Piot PHYS 571 Fall 2007 Direction of propagation The direction of the wave is given by k k perpendicular to E and B Let c be the angle between the velocity of the particle and k then From Velocity of light in the considered medium P Piot PHYS 571 Fall 2007 Shock wave feature Cerenkov radiation consists of a shock wave Effect similar to the Mach effect P Piot PHYS 571 Fall 2007 Shock wave feature I Cerenkov radiation consists of a shock wave Effect similar to the Mach effect P Piot PHYS 571 Fall 2007 Shock wave feature II The Shock wave feature inferred geometrically can be derived from the wave equation So A takes the same form as in vacuum under renormalization So we can directly write the potentials as P Piot PHYS 571 Fall 2007 Shock wave feature III Consider On another hand causality So Solve for t t P Piot PHYS 571 Fall 2007 Shock wave feature IV For Cerenkov radiation v cm to obtain t t real positive we need v 0 and v 2 v2 cm2 2 That is So Thus potentials and fields exist at time t only within a cone with apex lying at the present time position of the particle and apex angle P Piot PHYS 571 Fall 2007 Shock wave feature V The 4 potential is given by The denominator is Expliciting t t we gives P Piot PHYS 571 Fall 2007 Shock wave feature VI So finally the potentials are singularity In practice the singularity is smeared by the frequency dependence of which implies cm P Piot PHYS 571 Fall 2007