Radiation from accelerating charges We showed the Li nard Wiechert potential are P Piot PHYS 571 Fall 2007 Radiation from accelerating charges application In principle the field are easily obtained from the potentials using Problem here all the quantities have to be evaluated at a retarded time So we need to express in term of retarded quantities P Piot PHYS 571 Fall 2007 t as function of retarded quantities Consider On another hand the chain rule gives Which can be expressed from So The two highlighted equation result in P Piot PHYS 571 Fall 2007 as function of retarded quantities Consider Let be the gradient evaluated at t Then chains rule The two previous equations result in That is P Piot PHYS 571 Fall 2007 Electric field I In term of retarded quantities the E field is with We have So finally R P Piot PHYS 571 Fall 2007 Electric field II And finally Now let s consider This gives With and P Piot PHYS 571 Fall 2007 Electric field III Thus From the two latest highlighted equation we get Now we consider The t derivative of A is P Piot PHYS 571 Fall 2007 Electric field IV So the E field is finally given by Which simplifies to P Piot PHYS 571 Fall 2007 Electric field V Near field Velocity fields P Piot PHYS 571 Fall 2007 Far field Radiation fields Electric field of a uniformly moving charge I d dt 0 so Does this agree with what we learnt Yes Consider the drawing P Piot PHYS 571 Fall 2007 Electric field of a uniformly moving charge II Geometric considerations give P Piot PHYS 571 Fall 2007
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