Radiation from accelerating charges 1 We now concentrate in computing the radiated power by an accelerated charge From the Poynting vector definition the radiated power per unit of surface area is So From now on we will only consider radiation detected at very large R this is the far field approximation Only the second term that involves the acceleration is kept in the E B fields associated to the LienardWiechert potential P Piot PHYS 571 Fall 2007 Radiation from accelerating charges 2 From the instantaneous power radiated at time t per unit solid angle d is In the far field approximation we have Let consider n E P Piot PHYS 571 Fall 2007 Radiated power per unit of time So finally and P Piot PHYS 571 Fall 2007 Radiated energy per unit of time If we want to know dP t dt the power radiated per unit solid angle at time t it arrived at the enveloping surface then we must trace back to the proper retarded time Note also that so if a particle is suddenly accelerated for a time t an observer will see a light pulse at time t R c with a duration Energy is conserved total energy radiated total energy lost by the particle The fact that implied that energy radiated per unit of time is ret times the energy lost by the particle in the far field per unit of time P Piot PHYS 571 Fall 2007 Instantaneous rate of radiation 1 Hgf Work out the integrant From we get P Piot PHYS 571 Fall 2007 Instantaneous rate of radiation 2 So Recall and define the integrals So the power is P Piot PHYS 571 Fall 2007 Instantaneous rate of radiation 3 We finally have The last line is the relativistic generalization of Larmor s formula to recover the Larmor formula just take the limit 0 P Piot PHYS 571 Fall 2007 Power radiated in linear accelerators 1 In linear accelerators We need to evaluate the acceleration Start from the momentum Thus the radiated power is Lighter particles are subject to higher loss P Piot PHYS 571 Fall 2007 Power radiated in linear accelerators 2 One important question is how does the emission of radiation influence the charge particle dynamics The accelerator induce a momentum change of the form where we assumed the acceleration is along the z axis Let be the power associated to the external force The particle dynamics is affected when Pext is comparable to the radiated power P Piot PHYS 571 Fall 2007 Power radiated in linear accelerators 3 Consider a relativistic electron then So the effect seems to be negligible This is actually part of the story some coherent effect can kick in an induce some distortion when considering highly charged electron bunches for instance P Piot PHYS 571 Fall 2007 Power radiated in circular accelerators 1 Now and The radiated power is where E is the energy Let s introduce So radiative energy loss per turn is P Piot PHYS 571 Fall 2007 Power radiated in circular accelerators 2 That is For an e synchrotron this becomes Take E 1 TeV R 2 km we have Conclusion bad idea to build circular accelerator for HEP but good as copious radiation sources e g APS in Argonne P Piot PHYS 571 Fall 2007