P. Piot, PHYS 571 – Fall 2007Field 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 bλ<<1P. Piot, PHYS 571 – Fall 2007Field in the limit bλ>>1• We now consider the extreme case bλ>>1. The modified Bessel functions have the asymptotic expansion:• And the field associated to a uniformly moving charged particle takes the formP. Piot, PHYS 571 – Fall 2007Cerenkov condition• We saw that to get radiation (or energy loss) we needed either λ or εto be complex. • When investigating dielectric screening effects we considered the case ε ∈C• We now consider the case where λ is a pure imaginary number and ε∈R.•from •….P. Piot, PHYS 571 – Fall 2007Cerenkov condition• Consider the model for permittivityωεε=1ω=ω01/β2ωLω0P. Piot, PHYS 571 – Fall 2007Energy loss• Explicit the asymptotic form of the e.m. field in the energy loss equation. •givesP. Piot, PHYS 571 – Fall 2007Frank-Tamm energy loss formula•But λ∈I so λ∗/λ=-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 2007Direction of propagation• The direction of the wave is given by k, k perpendicular to E and B. Let θcbe the angle between the velocity of the particle and k then•From Velocity of light in the considered mediumP. Piot, PHYS 571 – Fall 2007Shock wave feature• Cerenkov radiation consists of a shock wave• Effect similar to the Mach effectP. Piot, PHYS 571 – Fall 2007Shock wave feature I• Cerenkov radiation consists of a shock wave• Effect similar to the Mach effectP. Piot, PHYS 571 – Fall 2007Shock 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 asP. Piot, PHYS 571 – Fall 2007Shock wave feature III• Consider• On another hand•So•Solve for (t-t’) :causalityP. Piot, PHYS 571 – Fall 2007Shock wave feature IV• For Cerenkov radiation (v>cm), to obtain t-t’ real positive we need ζ.v>0 and (ζ.v)2>(v2-cm2)ζ2. • 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 angleP. Piot, PHYS 571 – Fall 2007Shock wave feature V• The 4-potential is given by • The denominator is• Expliciting (t-t’) we givesP. Piot, PHYS 571 – Fall 2007Shock wave feature VI• So finally the potentials are• In practice the singularity is smeared by the frequency-dependence of ε(ω) which implies