HAPTER 6. TIME-VARYING FIELD AND MAXWELL EQUATION tatic electric charge tatic E an D E = 0 i D =ρ v N mutual relatinhip teay electric current tatic H an B H = J i B = 0 Accelerate charge Time-varying fiel E an H (time-varying current) Intercuple Frm thi pint time-varying quantitie are ente by cript r italic letter. E, H, ρ v an J, : time-varying electric fiel, magnetic fiel, vlume charge enity an current enity, repectively. Gau law, id = ρ an i B = 0 v, till hl. E an H are mifie t inclue the electrmagnetic inuctin an the iplacement current. Faraay experiment Maxwell hypthei Maxwell cmbine tw curl equatin fre an H Electrmagnetic wave prpagating with the ame velcity a light 6.1 FARADAY LAW Mving a magnet near a wire lp, r mving a wire lp near a magnet Electrmtive frce(emf), r a vltage Faraay law The inuce emf in a cle wire lp i equal t the negative time rate f change f the magnetic flux linkage with the lp. Φ emf = [ V ] (6-1) If the lp ha N turn f wire, Φ emf = N [ V ] (6-) Φ i the magnetic flux encle by a ingle turn f wire. Uing the electric fiel emf = E il (6-3) E i a functin f time in general ntant, if Φ / i cntant Time-Varying Fiel 6-1 Prprietary f Prf. Lee, Yen H
tatic E i cnervative le line integral f E i alway er E i crrepn t the ptential ifference E i a nn-cnervative fiel le line integral f E i emf E V mbine Eq. (6-) with Eq. (6-3), an ue urface integral f B fr Φ. Faraay law i then emf = E il = B i (6-4) N = 1 l an Minu ign : a urface bune by : the right-han rule : Len law The inuce emf pruce a current an therefre a magnetic flux In uch a irectin a t ppe the change in the flux linkage with the lp. V 1 > 0 in Fig. 6-1(a) V < in Fig. 6-1(b) 1 0 V 1 i an inuce vltage, nt an applie vltage that wul pruce an electric fiel in the ppite irectin t E in the lp. 6-1.1 Tranfrmer emf tatinary lp / t inie the integral B emf = E il = i (6-5) Time-Varying Fiel 6- Prprietary f Prf. Lee, Yen H
It i calle the tranfrmer emf Vltage inuce in a tatinary lp in B Applying tke therem t Eq. (6-5) B ( E) i = i r B E = (6-6) t Faraay law in pint frm, ne f the fur equatin f Maxwell. Example 6-1 Fin inuce E in a tatinary cnucting lp f raiu a, which i place in a patially unifrm time-varying magnetic flux enity, ( t) = B c ωta B. lutin Auming l t be clckwie, frm Eq. (6-5) B E il = i π ae φ = πa ( B cωt) =πa ωb in ωt Rearranging it ( ) E = E a = aωb in ωt a 1 φ φ φ Figure 6-3 plt the time interval in which B an E φ, nrmalie t unity, a functin f t. hae area inicate B i increaing an E φ therefre i given parallel a φ. Time-Varying Fiel 6-3 Prprietary f Prf. Lee, Yen H
6-1. Mtinal emf A lp mve in tatic B Mtinal emf Due t magnetic frce n free electrn f the cnuctr. emf = E il = Bi (6-7) i a part f the urface that i bune by, having nner B. epen n time in general. Referring t Fig. 6-4 B = B a in the regin y y Time-Varying Fiel 6-4 Prprietary f Prf. Lee, Yen H
Frm Eq. (6-7) emf = E il = Bi = ( Bxvt ) = Bxv where a an i cunterclckwie (6-8) emf lckwie current B in a irectin, pping t the increae in the flux alculatin f mtinal emf frm magnetic frce n free electrn Magnetic frce n an electrn F = e v B (6-9) m Time epenent Mtinal electric fiel intenity i efine by Fm Em = = v B (6-10) e The inuce emf emf = E il = ( v B) il (6-11) m Em l n the tp an bttm ie B = 0 n the left ie uring 0 < t < y / v emf x = 0 = v B a x B x v xi ax = (6-1) x= x An example mtinal emf referring t Fig. 6-5 liing bar mve n perfectly cnucting rail Time-Varying Fiel 6-5 Prprietary f Prf. Lee, Yen H
With the witch 1 pene. On the lp that cnit f bar, rail an gap between terminal 1 an l emf = E m il = ( v B) il u (6-13) x= l x= u = vb a i a x= vbx x x Perfectly cnucting rail are at equi-ptential V1 = vbx Frm the ther viewpint. Magnetic frce F m n charge + charge at l, - charge at u Infinite current in a x irectin? N current ue t E m Bth F m an E m frce electrn t mve in x -irectin in the bar. E m appear a an emf acr terminal 1 an. V = v B x (6-14) 1 Directin calculatin f mtinal emf frm Eq. (6-7) emf = Bi = B ( xy ) = Bx ( y + vt) a i a = Bxv (6-15) With the witch 1 cle Inuce emf urrent in clckwie irectin B in Pwer calculatin with witch 1 cle urrent I in the bar mving with v in B. Frm Eq. (5-15b) ( ) x= l I = = x= u y F I l B I B a x = IB x a y -irectin. Opping t the change in Φ. The mechanical pwer P = F iv = IB x v (6-16) m I Pwer iipate in the reitr R P = IV e 1 = IB x v (6-17) P m = P, the cnervatin f energy. e Time-Varying Fiel 6-6 Prprietary f Prf. Lee, Yen H
6-1.3 A Lp Mving in a Time-Varying Magnetic Fiel A cnucting lp mve in a time-varying magnetic fiel Ttal emf = tranfrmer emf + mtinal emf B emf = E il = i + i l ( v B) (6-18) Applying tke therem B E = + ( v B) (6-19) The intantaneu emf Φ emf = = B i (6-0) The time erivative may nt be taken inie the integral, becaue the bune urface may mve with time an cntribute t the change in the magnetic flux linkage with the lp. Example 6- A rectangular cnucting lp with ie a an b i in a time-varying magnetic flux enity B = B in ( ωt) a y. The lp rtate with an angular pee ω abut x -axi. The rtatin angle f the nrmal t the lp urface i ϕ =ωt α. Fin at time t (a) tranfrmer emf (b) mtinal emf (c) intantaneu emf lutin (a) Ttal magnetic flux encle by the lp at t Φ= B i = B in ( ωt) ay iaba = B in ( ω t) abc ϕ (6-1) Fixing ϕ t a cntant value, we btain the tranfrmer emf Φ emf = = Bωabc ( ωt ) c ϕ (6-a) Fr the given unit vectr a, accring t the right-han rule, the line integral f E hul be ne alng the lp paing the pint in the rer 1--3-4. The minu ign in Eq. (6- a) inicate that the terminal II i at the higher ptential, r the pitive terminal, at t = 0. Time-Varying Fiel 6-7 Prprietary f Prf. Lee, Yen H
(b) Mtinal emf, frm Eq. (6-11) emf = ( v B) il 1 = ( )( ) in ( ) ( ) 1 ω b a B ωt a y i x ax 4 1 + ω ( b )( ) in ( ) ( ) 3 a B ωt a y i xax = B ωabin ( ω t) in ϕ (6-b) The left an right ie cntribute n mtinal emf becaue cnuctr. v B i perpenicular t the (c) By ubtituting ( ωt α ) fr ϕ in Eq. (6-1), the intantaneu magnetic flux encle by the lp Φ= B in ( ωt) abc ( ωt α ) Intantaneu emf Φ emf = = Bωab c ( ωt ) c ( ωt α ) + Bωab in ( ωt ) in ( ωt α ) It i the ame a the um f the reult in Eq. (6-a) an (6-b). Time-Varying Fiel 6-8 Prprietary f Prf. Lee, Yen H
6- DIPLAEMENT URRENT DENITY Ampere circuital law, H = J (6-3) Divergence f bth ie i H = i J =0. =0, the cntinuity cnitin Vectr ientity The cntinuity cnitin fr time-varying fiel, i J ρ v / Eq. (6-3) i nt vali fr time-varying fiel = A parallel-plate capacitr A cle lp arun the lea Bune urface 1 r Ampere law t Fig. 6-7(a) H il = I : I penetrate 1 Ampere law t Fig. 6-7(b) H i l = 0 : N current penetrate An bviu cntraictin! Time-Varying Fiel 6-9 Prprietary f Prf. Lee, Yen H
A current i nt the nly urce f a time-varying magnetic fiel An increae r ecreae I Magnetic fiel between electre Electric fiel between electre E = Q a ε A (6-4) Q A ε : ttal charge n a cnucting plate : area f the plate : permittivity f free pace. Time erivative f bth ie, with I = Q / ( εe ) I = a : a current enity (6-5) A The iplacement current enity i efine by D D t J (6-6) The generalie Ampere law D = + t H J (6-7) The generalie Ampere law in integral frm D il = + i H J (6-8) The current enity J inclue bth the cnuctin an cnvectin current heck the generalie Ampere law Divergence f bth ie f Eq. (6-7) i H = ij + ( id ) Uing Gau law i H = ij + ( ρ v ) : alway true t =0. =0, the cntinuity cnitin A vectr ientity Example 6-3 An ac vltage V = V in ω t i applie acr an air-gap parallel-plate capacitr a hwn in Fig. 6-8. Ignring ege effect, (a) hw that the iplacement current i equal t the cnuctin current. (b) fin the magnetic fiel arun the cnuctr lea an in the gap. Time-Varying Fiel 6-10 Prprietary f Prf. Lee, Yen H
lutin (a) Fr electre eparate by V D =ε E =ε in ωt Diplacement current enity J D D V = = ωε c ωt Diplacement current in the capacitr f = ε / V ID = J D i = Aωε c ωt = V ωc ωt nuctin current in the lea V I = = Vωc ω t We have I D = I A (b) We firt apply the generalie Ampere law t a cle lp arun the capacitr lea. The iplacement current i er in the capacitr lea, becaue f n electric fiel in a perfect cnuctr. il = I H 1 πρ H I 1φ = r 1 I H (6-9) = a φ πρ The cnuctin current i er in the gap f the capacitr H il = J D i (6-30) Fr the capacitr plate f raiu a, uing I D = I J I D I D = πa πa (6-31) Inerting Eq. (6-31) in Eq. (6-30) I πρ πρ H φ = πa Thu ρ H = I a φ ( a ) πa I = a φ πρ ρ (6-3a) H ( ρ a ) (6-3b) At a far raial itance ρ> a, the magnetic fiel i cntinuu alng the axial irectin; the magnetic fiel in Eq. (6-9) an (6-3b) are ientical. Time-Varying Fiel 6-11 Prprietary f Prf. Lee, Yen H
6-3 MAXWELL EQUATION Unificatin f tw therie Electricity Magnetim Diplacement current enity Differential wave equatin Exitence f electrmagnetic wave (ame velcity a light) Maxwell equatin B E = t (6-33a) D H = J + t (6-33b) id ρ (6-33c) = v i B = 0 (6-33) Any electrmagnetic fiel mut atify the fur equatin ntitutive equatin D =εe (6-34a) B =μh (6-34b) nuctin an cnvectin current enitie J =σe (6-35a) J ρ (6-35b) = ve Time-Varying Fiel 6-1 Prprietary f Prf. Lee, Yen H
Lrent frce equatin F = q ( E + v B) (6-36) ntinuity equatin ρv i J = t (6-37) Example 6-4 hw that the time-varying fiel E = E a c ( kx ωt) an = E a c ( kx ωt) where k =ω με, atify Maxwell equatin in free pace. lutin ubtituting E an H in Eq. (6-33a) ε E ay c ( kx ω t) = Eμ a c ( kx ωt) μ r ke a in ( kx ω t ) = ωe με a in ( kx ωt ) The equatin i true fr k =ω με. ubtituting the fiel in Eq. (6-33b), with J = 0 in free pace ε E a c ( kx ω t) = ε E y c ( kx ω t) μ a r ε ke y in kx t E y kx t μ a a ( ω ) =ωε in ( ω ) The equatin i true fr k =ω μ ε. ubtituting the fiel in Eq. (6-33c) id = i εe ay c ( kx ω t) = 0 Gau law fr D i atifie in free pace f ρ = v 0. ubtituting the fiel in Eq. (6-33) ε i μe a c ( kx ω t) = 0 μ Gau law fr B i atifie. y H ε μ, In the ame way, it can be hwn that the time-varying fiel E = E a c ( kx +ωt) an H = E a c ( kx +ωt) al atify Maxwell equatin. ε μ y Time-Varying Fiel 6-13 Prprietary f Prf. Lee, Yen H
6-3.1 Maxwell Equatin in Integral Frm The pint frm i ueful in ecribing the lcal effect A lcal cnequence berve at a pint in pace i irectly relate t a lcal urce at the pint. The integral frm i ueful in ecribing the nnlcal effect A lcal cnequence at a pint i relate t a urce itribute in the neighbrh the pint. The integral frm i eaier t unertan, becaue the tep f integratin, invlving increment, pruct, ummatin, an n, may be mre realitic than the ifferentiatin. The pint frm can be cnverte t the integral frm by the ue f ivergence an tke therem, r vice vera. Maxwell equatin in integral frm B E il = i (6-38a) D H il = J + i (6-38b) D i = ρ v v (6-38c) V B i = 0 (6-38) The pitive irectin f i relate t the irectin f by the right-han rule. 6-3. Electrmagnetic Bunary nitin Bunary cnitin fr E an H E = E (6-39a) 1t t H H = J (6-39b) 1t t n J n i the urface current enity nrmal t H 1t an H t. The bunary cnitin fr D an B D D = ρ (6-39c) 1n n B = B (6-39) 1n n ρ i the free urface charge enity. Time-Varying Fiel 6-14 Prprietary f Prf. Lee, Yen H
At the interface between tw lle ielectric ρ = J = 0 E1 t = E t (6-40a) H1 t = H t (6-40b) D1 n = D n (6-40c) B = B (6-40) 1n n At the interface between a lle ielectric(meium 1) an a perfect cnuctr(meium ) E = H = D = B = 0 in the cnuctr E (6-41a) H (6-41b) D ρ (6-41c) B (6-41) 1t = 0 1t = J 1n = = 1n 0 an E t = H t = 0 (6-4a) D = B = (6-4b) n n 0 ρ an J are the irect urce f tatic fiel D an H. ρ an J may nt be the irect urce f time-varying fiel. An electrmagnetic wave impinging n a perfect cnuctr The incient an reflecte wave inuce time-varying ρ an n the cnuctr urface. J Time-Varying Fiel 6-15 Prprietary f Prf. Lee, Yen H
6-4 RETARDED POTENTIAL calar electric ptential V frm a tatic vlume charge enity ρ v. Vectr magnetic ptential A frm a teay current enity J. V = A = V ' V ' ρ v v ' 4πεR μjv ' 4πR (6-43a) (6-43b) Fr time-varying ρ v an J, they becme the retare calar ptential r the retare vectr ptential. i B = 0 till hl fr the time-varying fiel Time-varying vectr magnetic ptential i efine by B = A (6-44) Inerting Eq. (6-44) in Faraay law B E = t = ( A) Rewriting it A E + = 0 We efine the time-varying calar ptential V A E + V r A E = V t (6-45) (6-46) (6-47) (6-48) It can reuce t the previu relatin, E = V, in the teay ytem By ubtituting Eq. (6-44) in the generalie Ampere law D A =μ J +μ (6-49) t Inerting Eq. (6-48) in Eq. (6-49) an uing = ( i ) με = μ + i +με A A A A V A J A (6-50) Lrent cnitin fr ptential V i A = με (6-51) t It reuce t i A = 0 in the teay ytem. Time-Varying Fiel 6-16 Prprietary f Prf. Lee, Yen H
Frm Eq. (6-50) an (6-51), the inhmgeneu wave equatin fr vectr ptential A A με = μj (6-5) It reuce t vectr Pin equatin a Eq. (5-53) in the teay ytem. The lutin i given by a traveling wave f velcity 1/ με. Rewriting Gau law uing Eq. (6-48) ρv ie = ε A = i V = V ( ia) (6-53) Uing the Lrent cnitin, we btain the inhmgeneu wave equatin fr calar ptential V ρv V με = (6-54) t ε The lutin f the inhmgeneu ifferential wave equatin μj ( t R / v) A ( r, t) = v' (6-55) V ' 4πR ρv ( t R / v) v' V ( r, t) = (6-56) V ' 4πεR which are calle the retare vectr ptential an the retare calar ptential. At time t, at a itance R frm the urce. A an V are etermine by the value f J r ρ v given at an early time ( t R / v). Time elay R /v i the time taken by the wave uring traveling the itance R. Example 6-5 Fin retare vectr ptential at a pint p ue t an infiniteimal current element lcate at the rigin, carrying a current I = I c ω t. lutin Rewriting Eq. (6-55) μj ( t R / v) μi ( t R / v) A ( r, t) = v' = ' V ' 4 L' l π a (6-57) R 4πR Fr an infiniteimally mall current element f height h, we aume R t be cntant. Then, Eq. (6-57) reuce t μh A = I c ω( t R / v ) 4πR a The vectr ptential at a itance R i retare by / R v with repect t I at the rigin. The retaratin i ue t the traveling time f the electrmagnetic wave frm the rigin t p. Time-Varying Fiel 6-17 Prprietary f Prf. Lee, Yen H
Time-Varying Fiel 6-18 Prprietary f Prf. Lee, Yen H