P. Piot, PHYS 571 – Fall 2007Radiation from accelerating charges• Radiation emitted by particle attime t’ reach observer at time t. • Retardation occurs due to the finitevalue of speed of light• Position of the particle at time t not relevant, need to know the particle history• Mark your calendar 11/30: Dr. Rui Li of Jefferson Laboratory will discuss retardation effect in synchrotron radiation and the inherent associated problem in charged particle beams.P. Piot, PHYS 571 – Fall 2007• Synchrotron radiation• Undulator radiation•Smith Purcell• Transition radiationRadiation from accelerating charges: applicationP. Piot, PHYS 571 – Fall 2007Radiation from accelerating charges: problems…• Wakefield• Coherent and Incoherent synchrotron radiation3/120)24(ρσzL =Overtaking length50 100 150 200 250 300 350 40020406080100120140160180200P. Piot, PHYS 571 – Fall 20074-potential associated to a moving charge I• Start with Maxwell’s equation (inhomogeneous)• Consider the Lorenz’ gauge• Which can be rewritten• Solution of the equation is in term of Green’s functionIf free space:P. Piot, PHYS 571 – Fall 20074-potential associated to a moving charge II• With previous eq. rewrites • Solve using Fourier transform: define• In Fourier space the d’Alembert equation reduces towhereP. Piot, PHYS 571 – Fall 2007• The Green function is obtained by inverse Fourier transform:• Consider the integral of k04-potential associated to a moving charge IIIpoles ±κk0P. Piot, PHYS 571 – Fall 2007•SoD becomes• Introducing the angular part can be integrated:•..4-potential associated to a moving charge IVP. Piot, PHYS 571 – Fall 20074-potential associated to a moving charge V•……• Using the identity• D becomesP. Piot, PHYS 571 – Fall 20074-potential associated to a moving charge VI• And the retarded potential is• Retarded and advanced…P. Piot, PHYS 571 – Fall 2007Liénard-Wiechert I• And the retarded potential is• The 4-current•Expliciting• Or, using ,P. Piot, PHYS 571 – Fall 2007Liénard-Wiechert II• So the 4-potential takes the form• Which can be