Today in Physis 8: eleti dipole adiation II The fa field Veto potential fo an osillating eleti dipole Radiated fields and intensity fo an osillating eleti dipole Total satteing oss setion of a dieleti sphee Rada satteing oss setion, in squae metes at fequeny 0 GHz, of some objets odinaily obseved by ada systems Fom Antennas, by JD Kaus and R Mahefka 5 Mah 004 Physis 8, Sping 004 Fa-field adiation fom an osillating eleti dipole Last time we found that the sala potential fo an osillating eleti dipole is p0 osθ osθ V = osω t sin ω t, whee p0 = q0 d, and q0 is the amplitude of the two sepaated eleti hages Thee ae at least two inteesting limits in whih to view this solution: DC (ω = 0) Then we get a familia esult: p0 osθ VDC =, the same as we got in PHY 7 fo a stati eleti dipole (http://wwwpasohesteedu/~dmw/phy7/letues/let_0bpdf) 5 Mah 004 Physis 8, Sping 004 Fa-field adiation fom an osillating eleti dipole High fequenies and lage In this ase, Fa field ω = λ π, (o adiation zone ) so that all told we have assumed λ π d That makes the seond tem in V muh bigge than the fist: osθ Vad (, θ, t) = sin ω t Meanwhile, the uent in the wie that onnets the two haged onduting balls is dq I = = q0ωsin ωt, dt 5 Mah 004 Physis 8, Sping 004 3 () Univesity of Roheste
Veto potential fo an osillating eleti dipole so the etaded veto potential, in the fa field, is d Id q0ωsinω( t ) zˆ dz Aad = = d The ange of integation isn t muh ompaed to the distanes involved in the integand Conside, theefoe, a F( x) dx = f ( a) f ( a) fo small a ( ): a df df = f ( 0 ) + a+ f ( 0 ) + ( a) + dx x= 0 dx x= 0 df Note that z = 0 a = af( 0 ) dx means = x= 0 5 Mah 004 Physis 8, Sping 004 4 Veto potential fo an osillating eleti dipole Thus q0ωsinω( t ) zˆ sinω( t ) Aad = d = zˆ Let s onvet this to spheial oodinates, to math the expession fo V: sinω( t ) ˆ Aad = ( ˆ θ sinθ + ˆ osθ ) sinθ ẑ sinθsinω( t ) θ = ˆ θ osθ osθsinω( t ) ˆ θˆ Fom hee it s a staightfowad matte to get the fields: 5 Mah 004 Physis 8, Sping 004 5 Radiated fields and intensity fo an osillating eleti dipole A V V ˆ A E = V = ˆ θ t θ t osθ osθ = sinω t os t ˆ ω sinθ + sinω t ˆ θ sinθ ˆ osθ + osω t + osω t ˆ θ sinθ osω t ˆ θ 5 Mah 004 Physis 8, Sping 004 6 () Univesity of Roheste
Radiated fields and intensity fo an osillating eleti dipole Similaly, B = A= ( A ) A ˆ θ θ φ sinω ( t ) = sinθ ( sinω( t ) ) osθ ˆ φ θ ω sinθ sinω( t ) = sinθ osω( t ) + ˆ φ ω 0 sin sin os ( ) ˆ ˆ p ω θ θ ω t = os ω ( t ) φ φ 5 Mah 004 Physis 8, Sping 004 7 Radiated fields and intensity fo an osillating eleti dipole Sine k = ω, p0 = p0zˆ, and ˆ zˆ = ˆ φ, we an wite a moe ompat equivalent to these fomulas: sinθ B = k ( ˆ p0 ) os ω t, E = B ˆ Both E and B ae pependiula to ˆ Sine the osω ( t ) fato makes this a wave that tavels towad +, the fa-field adiation fom an osillating eleti dipole is a tansvese, expanding, spheial wave 5 Mah 004 Physis 8, Sping 004 8 Radiated fields and intensity fo an osillating eleti dipole The Poynting veto fo these adiated fields omes out athe simply: sinθ S= E B = os ω ( t ) ˆ θ ˆ φ 4π 4π Noting that ˆ θ ˆ φ= ˆ, and ealling that os ( ωt δ) =, this gives us S sinθ = ˆ = Iˆ 8π 5 Mah 004 Physis 8, Sping 004 9 () Univesity of Roheste 3
Radiated fields and intensity fo an osillating eleti dipole Contou map of the magnitude of the Poynting veto as a funtion of time, fo an osillating eleti dipole Animation by Akia Hiose, U Saskathewan Pof Hiose s animations an be found at: http://physisusaska/~hiose/ep5/adiationhtm 5 Mah 004 Physis 8, Sping 004 0 Radiated fields and intensity fo an osillating eleti dipole A onvenient way to envision the 3-D distibution of this adiated intensity is to plot, in θ pola oodinates, the intensity 05 fo a given distane Beause of the sin θ fato, sin θ 05 0 05 the intensity is lagest fo ˆ pointing towad θ = π, 05 and is zeo along the z axis I sin θ In between it looks like the plot on the ight 5 Mah 004 Physis 8, Sping 004 Radiated fields and intensity fo an osillating eleti dipole Fo example: the intensity tavelling at θ = 30 (sinθ = ) is half the intensity at θ = 45 (sinθ = ), 05 though in eah ase the dietion is adially θ sin θ outwad, and the fields ae 05 0 05 still tansvese 05 (pependiula to ˆ ) I sin θ 5 Mah 004 Physis 8, Sping 004 () Univesity of Roheste 4
Radiated fields and intensity fo an osillating eleti dipole These kinds of pola plots, that show whih way the adiated intensity is geatest, and by how muh, ae often used to desibe the dietivity of antennas moe ompliated than the dipole Hee, fo example, is what you d get fo ten dipoles in a ow, spaed λ/4 apat with phase shifts of π/ between onseutive uents (fom Kaus, Antennas, 3 d edition) 5 Mah 004 Physis 8, Sping 004 3 Total satteing oss setion of a dieleti sphee The total powe emitted by a dipole is omputed by integating the Poynting veto ove any sufae that enloses the dipole, like any sphee fo whih the adius puts it in the fa field: 4 sin θ P = ˆ ˆ S da= 3 sinθdθdφ 8π 4 π π 4 4 3 4 = dφ dθ sin θ π 3 = = 3 3 3 8π 8π 3 0 0 It doesn t matte how this dipole is made to osillate; one it does, though, it will adiate as desibed, with the total powe given above 5 Mah 004 Physis 8, Sping 004 4 Total satteing oss setion of a dieleti sphee Suppose, instead of the dipole we stated with, we have a dipole indued by an osillating extenal eleti field E0 osωt fo example, a small dieleti sphee, with adius a small enough to onside that thee is no delay in popagation of the field aoss it The dipole moment of suh a sphee is 4π 3 4π 3 p = a P = a χee0 osωt p0 os ωt 3 3 Suppose futhemoe that the extenal eleti field is supplied by a plane eletomagneti wave that s passing by the ube The powe adiated by the dipole is thus taeable bak to the inident wave: the ube has satteed some of the inident light, due to adiation by the indued dipole moment 5 Mah 004 Physis 8, Sping 004 5 () Univesity of Roheste 5
Satteing of eletomagneti plane-wave powe by a dieleti sphee 5 Mah 004 Physis 8, Sping 004 6 Total satteing oss setion of a dieleti sphee It is ustomay to elate the total powe in satteed light to the intensity of inident light by P = σ I satteed s I, whee σ s is the total satteing oss setion of ou dieleti sphee σ s has units of aea; one an think of it as something like the aea of the shadow ast by the dieleti sphee Note, howeve, that in geneal this aea an be vey diffeent fom the geometial oss-setional aea of the objet that does the satteing 5 Mah 004 Physis 8, Sping 004 7 Total satteing oss setion of a dieleti sphee In the pesent ase of the dieleti sphee, 6 4 4π a χe ω P = E 3 0 = σsii = σs E0 3 3 8π 3 6 4 4π a χe ω σs = 3 4, o Note that it s not π a Note espeially the stong dependene on angula fequeny This will be the uial point of ou disussion next letue 5 Mah 004 Physis 8, Sping 004 8 () Univesity of Roheste 6