11 Radiation in Non-relativistic Systems

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1 Radiation in Non-relativisti Systems. Basi equations This first setion will NOT make a non-relativisti approximation, but will examine the far field limit. (a) We wrote down the wave equations in the ovariant gauge: The gauge ondition reads (b) Then we used the green funtion of the wave equation to determine the potentials (Φ, A) Φ(t, r) = A(t, r) = Here T (t, r) is the retarded time Φ =ρ(t o, ) (.) A =J(t o, )/ (.2) tφ + A = 0 (.3) G(t, r t o ) = 4π r δ(t t o + r ) (.4) d 3 x o 4π r ρ(t, ) (.5) d 3 x o 4π r J(T, )/ (.6) T (t, r) = t r (.7) () We used the potentials to determine the eletri and magneti fields. Eletri and magneti fields in the far field are A (t, r) = J(T, ) (.8) 4πr and In the far field (large distane limit r ) limit we have B(t, r) = n ta (.9) E(t, r) =n n ta = n B(t, r) (.0) T = t r + n ro (.) And we reording the derivatives t (.2) T r ( ) o = + n (.3) T T = t 45

2 46 CHAPTER. RADIATION IN NON-RELATIVISTIC SYSTEMS (d) We see that the iation (eletri field) is proportional to the transverse piee of the t J n (n t J) = t J n(n t J) (.4) In general the transverse projetion of a vetor is n (n V ) = V n(n V ) (.5) (e) Power iated per solid angle is for r is and dw dp (t) = = energy pebservation time per solid angle (.6) dtdω dω dp (t) dω =r2 S n (.7) = re 2 (.8).2 Examples of Non-relativisti Radiation: L3 In this setion we will derive several examples of iation in non-relativisti systems. In a non-relativisti approximation T = t r + n (.9) small The underlined terms are small: If the typial time and size sales of the soure are T typ and L typ, then t T typ, and L typ, and the ratio the underlined term to the leading term is: L typ T typ (.20) This is the non-relativisti approximation. For a harmoni time dependene, /T typ ω typ, and this says that the wave number k = 2π λ is small ompared to the size of the soure, i.e. the wave length of the emitted light is long ompared to the size of the system in non-relativisti motion: 2πL typ λ (.2) (a) Keeping only t r/ and dropping all powers of n / in T results in the eletri dipole approximation, and also the Larmour formula. (b) Keeping the first order terms in n (.22) results in the magneti dipole and quadrupole approximations. The Larmour Formula (a) For a partile moves slowly with veloity and aeleration, v(t) and a(t) along a trajetory r (t) (b) We make an ultimate non-relativisti approximation for T Then we derived the iation field by substituting the urrent into the Eqs. (.8),(.9), and (.7) for the iated power T t r t e (.23) J(t e ) = ev(t e )δ 3 ( r (t e )) (.24)

3 .2. EXAMPLES OF NON-RELATIVISTIC RADIATION: L3 47 () The eletri field is Notie that the eletri field is of order E = e 4πr 2 n n a(t e) (.25) E e 4πr (d) The power per solid angle emitted by aeleration at time t e is Notie that the power is of order a(t e ) 2 (.26) dp (t e ) dω = e2 (4π) 2 3 a2 (t e ) sin 2 θ (.27) P re 2 a2 3 (.28) (e) The total energy that is emitted is P (t e ) = e2 2 a 2 (t e ) 4π 3 3 (.29) The Eletri Dipole approximation (a) We make the ultimate non-relativisti approximation J(t r + n ) J(t r ) (.30) Leading to an expression for A A = 4πr tp(t e ) (.3) where the dipole moment is p(t e ) = d 3 x o ρ(t e ) (.32) (b) The eletri and magneti fields are E =n n ta (.33) = 4πr 2 n n p(t e) (.34) B =n E (.35) () The power iated is dp (t e ) dω = 6π 2 p 2 (t e ) 3 sin 2 θ (.36) (d) For a harmoni soure p(t e ) = p o e iω(t r/) the time averaged power is P = ω 4 4π 3 3 p o 2 (.37)

4 48 CHAPTER. RADIATION IN NON-RELATIVISTIC SYSTEMS The magneti dipole and quadrupole approximation: L32 (a) In the magneti dipole and quadrupole approximation we expand the urrent J(T ) J(t e ) + n t J(t e, )/ } {{} eletri dipole next term (.38) The next term when substituted into Eq. (.8) gives rise two new ontributions to A, the magneti dipole and eletri quadrupole terms: A = (b) The magneti dipole ontribution gives where m is the magneti dipole moment. A E eletri dipole + A M mag dipole + A E2 eletri-quad (.39) A M = n 4πr ṁ(t e) (.40) m 2 J(t e, )/, (.4) () The struture of magneti dipole iation is very similar to eletri dipole iation with the duality transformation (d) The power is E-dipole M-dipole (.42) p m (.43) E B (.44) B E (.45) dp M (t e ) dω = m2 sin 2 θ 6π 2 3 (.46) (e) The power iated in magneti dipole iation is smaller than the power iated in eletri dipole iation by a fatof the typial veloity, v typ squared: P M P E m2 p 2 ( vtyp ) 2 (.47) where v typ L typ /T typ Quadrupole rdiation (a) For quadrupole iation we have A j,e2 = n i 24πr Q ij 2 (.48) where Q ij is the symmetri traeless quadrupole tensor. Q ij = d 3 x o ρ(t e, ) ( 3ror i o j roδ 2 ij) (.49) (b) The eletri field is E = [... 24πr 3 Q n n(n Q... n) ] (.50)... where... (more preisely) the first term in square brakets means n i Q ij, while the seond term means, (n l Q lm n m )n j.

5 .3. ATTENAS 49 () A fair bit of algebra shows that the total power iated from a quadrupole form is ab P = 720π 5 Q Q ab (.5) (d) For harmoni fields, Q = Q o e iωt, the time averaged power is rises as ω 6 P = ( ω ) 6 Q 2 440π o (.52) (e) The total power iated iated in quadrupole iation to eletri-dipole iation for a typial soure size L typ is smaller: P E2 ( ) 2 P E ωltyp (.53).3 Attenas (a) In an antenna with sinusoidal frequeny we have (b) Then the iation field for a sinusoidal urrent is: J(T, ) = e iω(t r + n ro ) J( ) (.54) A = e iω(t r/) 4πr e iω In general one will need to do this integral to determine the iation field. n ro J( )/ (.55) () The typial iation resistane assoiated with driving a urrent whih will iate over a wide range of frequenies is R vauum = µ o = µ o /ɛ o = 376 Ohm.