dP d dt dP 1 Frequency dependence of radiation The angular distribution of radiation, the power emitted per sold angle is = r2S . f, where S is the Poynting vector. S(r, t) = r E(r, t). Er, t) for observer's po- sitions and times (r, t) in the radiation zone. The aim of this problem is to understand the frequency dependence of any emitted radiation. First, recall the Fourier transform, for example E(r, t) di Er,w)e-iwt. The inverse is effected using the delta function:8(w W') = L teilwwt, that E(rw) E(r, t)eiwt. The total energy radiated per solid angle date = L dt? (a) Show that are = r coc | dw Er,w). E*(r,w). (b) Show that in the Lorentz gauge the frequency-dependent vector potential Ar,w) can be ex- pressed in the radiation zone as Ar,w) = heikr J(k,w), where k = w/c and J(kw) = dr'dt'/ V2 J(r', t')e reiwt', where k = kr. (c) One and define write The angular distribution of energy per may solid angle per frequency. Show that I 100 [kJ(k.w).J*(kw) - (k. J(kw))(k.J*(kw))]. ikor du d92 po dw- d2U dwo22 dl d22 dw821 dP d dt dP 1 Frequency dependence of radiation The angular distribution of radiation, the power emitted per sold angle is = r2S . f, where S is the Poynting vector. S(r, t) = r E(r, t). Er, t) for observer's po- sitions and times (r, t) in the radiation zone. The aim of this problem is to understand the frequency dependence of any emitted radiation. First, recall the Fourier transform, for example E(r, t) di Er,w)e-iwt. The inverse is effected using the delta function:8(w W') = L teilwwt, that E(rw) E(r, t)eiwt. The total energy radiated per solid angle date = L dt? (a) Show that are = r coc | dw Er,w). E*(r,w). (b) Show that in the Lorentz gauge the frequency-dependent vector potential Ar,w) can be ex- pressed in the radiation zone as Ar,w) = heikr J(k,w), where k = w/c and J(kw) = dr'dt'/ V2 J(r', t')e reiwt', where k = kr. (c) One and define write The angular distribution of energy per may solid angle per frequency. Show that I 100 [kJ(k.w).J*(kw) - (k. J(kw))(k.J*(kw))]. ikor du d92 po dw- d2U dwo22 dl d22 dw821