Pwer Flw in Electrmagnetic Waves Electrmagnetic Fields The time-dependent pwer flw density f an electrmagnetic wave is given by the instantaneus Pynting vectr P t E t H t ( ) = ( ) ( ) Fr time-varying fields it is imprtant t cnsider the time-average pwer flw density 1 T 1 T Pt () = Pt () dt= Et () Ht () dt T 0 T 0 where T is the perid f bservatin. Amangawa, 006 Digital Maestr Series 89
Cnsider time-harmnic fields represented in terms f their phasrs E( t) = Re{ E exp( jω t) } = Re{E} csωt Im{E} sinωt Ht ( ) = Re H exp( jω t) = Re{H} csωt Im{H} sinωt { } The time-dependent Pynting vectr can be expressed as the sum f the crss-prducts f the cmpnents Et ( ) Ht ( ) = Re{E} Re{H} cs ωt + Im{E} Im{H} sin ωt Re{E} Im{H} + Im{E} Re{H} csωtsin ωt ( ) csωt sinω t= sin ω t ) (Nte that: 1 Amangawa, 006 Digital Maestr Series 90
The time-average pwer flw density can be btained by integrating the previus result ver a perid f scillatin T. The pre-factrs cntaining field phasrs d nt depend n time, therefre we have t slve fr the fllwing integrals: 1 T 1 t sinωt cs ω tdt= T 0 T + = 4ω 1 T 1 t sinωt sin ω tdt= T 0 T = 4ω T 0 T 0 1 1 1 T 1 sin ωt csωt sin ω t dt= T 0 = T ω T 0 0 Amangawa, 006 Digital Maestr Series 91
The final result fr the time-average pwer flw density is given by 1 T Pt () = Et () Ht () dt T 0 1 = Re{E} Re{H} + Im{E} Im{H} ( ) Nw, cnsider the fllwing crss prduct f phasr vectrs * E H = Re{E} Re{H} + Im{E} Im{H} + j ( Im{E} Re{H} Re{E} Im{H} ) Amangawa, 006 Digital Maestr Series 9
By cmbining the previus results, ne can btain the fllwing time average rule 1 T 1 Pt () = Et () Ht () dt= Re E H T 0 We als call cmplex Pynting vectr the quantity 1 P= E H { * } NOTE: the cmplex Pynting vectr is nt the phasr f the timedependent pwer nr that f the time-average pwer density! Pt () = Re P Pt () = Re Pexp( jωt) Phasr ntatin cannt be applied t the prduct f tw timeharmnic functins (e.g., P( t )), even if they have same frequency. * { } dn't try { } ( ) Amangawa, 006 Digital Maestr Series 93
Cnsider a 1-D electr-magnetic wave mving alng the z-directin, with a specified electric field amplitude E E x( z) = Eexp( αz)exp( jβz) E H ( ) y z = exp( αz)exp( jβz)exp( jτ) η The time-average pwer flw density is αz αz e e Re Pwer in a lssy medium decays as exp(-α z)! * z j z E z j z j 1 { * } 1 α β α β τ Pt ( ) = Re E H = Re Ee e e e e η 1 { jτ} 1 = E e = E cs τ η η Amangawa, 006 Digital Maestr Series 94
Cnsider the same wave, with a specified amplitude fr the magnetic field H ( z) = H exp( αz)exp( jβz) y E ( z) = η H exp( αz)exp( jβz)exp( jτ) x The time-average pwer flw density is expressed as α β α β τ 1 Pt () = Re η He e He e e 1 αz = η H e cs τ { z j z * z j z j } If α is the attenuatin cnstant fr the electrmagnetic fields α is the attenuatin cnstant fr pwer flw. Amangawa, 006 Digital Maestr Series 95
If the wave is generated by an infinitesimally thin sheet f unifrm current J s (embedded in an infinite material with cnductivity σ) we have fr prpagatin alng the psitive z-directin (nrmal t the plane f the current sheet):i Js J H = E = η J z Pt () s α = ηe csτ 8 s Fr this ideal case, an identical wave exists, prpagating alng the negative z-directin and carrying the same amunt f pwer. Amangawa, 006 Digital Maestr Series 96
Pynting Therem Cnsider the divergence f the time-dependent pwer flw density Pt () = Et () Ht () = Ht () Et () Et () Ht () ( ) The curls can be expressed by using Maxwell s equatins H E Pt () = µ Ht () σet () Et () εet () t t 1 1 = σe () t ε E () t µ H () t t t Density f dissipated pwer Rate f change f stred electric energy density Rate f change f stred magnetic energy density This is the differential frm f Pynting Therem. Amangawa, 006 Digital Maestr Series 97
Nw, integrate the divergence f the time-dependent pwer ver a specified vlume V t btain the integral frm f Pynting therem V Pt () dv = Pt () ds = S Pwer Flux thrugh S 1 1 = σe () t dv εe () t dv µ H () t dv t t V V V Pwer dissipated in vlume Rate f change f electric energy stred in vlume Rate f change f magnetic energy stred in vlume Amangawa, 006 Digital Maestr Series 98
Typical applicatins P in () t α =? P ut () t 1 m L Watts Put () t = Pin ()exp( t αl) m 1 Put () t Nepers α= ln L Pin( t) m Amangawa, 006 Digital Maestr Series 99
Example: Watts Watts Pin() t = 30 ; P () 5 ; 0 m ut t = L = m m Nepers α = 0.0448 m Pay attentin t the lgarithms: () () ln Put t Pin t = ln Pin() t Put () t Amangawa, 006 Digital Maestr Series 100
SURFACE A SURFACE B A P in ( t) = Pwer IN B P ut ( t) = Pwer OUT Area = Area(A) = Area(B) Pwer IN = Pt ( ) ds= Pt ( ) Area A Pwer OUT = Pt ( ) ds= Pt ( ) Area Pt () = Pt () exp( αl) B B A A Pwer dissipated between A and B? Pwer dissipated = Pwer IN Pwer OUT B L A B Amangawa, 006 Digital Maestr Series 101
Example Area = 5 m 8.44637 General Lssy medium Electrmagnetic Fields ; L= 1.0 cm; f = 1.0 GHz; E = 10 V/m ε=ε ; µ=µ ; σ= 0.45755 S/m σ = ωε η= 130.88 0.75rad = 130.88 41.534 α = 40.0 Ne/m; Pin( t) = 0.86 W/m ; P () t = P () t exp( α L) = 0.1845 W/m ; ut B in A Pwer IN = Area Pin( t) Pwer OUT = Area Pt ( ) 1.43 W 0.643 W Pwer dissipate d = Pwer IN Pwer OUT = B = = 0.7876 W Amangawa, 006 Digital Maestr Series 10