P. Piot, PHYS 571 – Fall 2007Radiation from accelerating charges• We showed the Liénard Wiechert potential areP. Piot, PHYS 571 – Fall 2007• 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. Radiation from accelerating charges: applicationP. 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 inP. 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 isP. Piot, PHYS 571 – Fall 2007Electric field I• In term of retarded quantities the E-field is•with• We have• So finally)RP. Piot, PHYS 571 – Fall 2007Electric field II•And finally• Now let’s consider• This gives•With • andP. Piot, PHYS 571 – Fall 2007Electric 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 2007Electric field IV• So the E- field is finally given by • Which simplifies toP. Piot, PHYS 571 – Fall 2007Electric field V• /………..Near field Velocity fieldsFar field Radiation fieldsP. Piot, PHYS 571 – Fall 2007Electric field of a uniformly moving charge I•dβ/dt =0 so • Does this agree with what we learnt?• Yes! Consider the drawing:P. Piot, PHYS 571 – Fall 2007Electric field of a uniformly moving charge II• Geometric considerations
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