# TIME VARYING MAGNETIC FIELDS AND MAXWELL S EQUATIONS

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1 TIME VARYING MAGNETIC FIED AND MAXWE EQUATION Introuction Electrosttic fiels re usull prouce b sttic electric chrges wheres mgnetosttic fiels re ue to motion of electric chrges with uniform velocit (irect current) or sttic mgnetic chrges (mgnetic poles); time-vring fiels or wves re usull ue to ccelerte chrges or time-vring current. ttionr chrges Electrosttic fiels te current Mgnetosttic fiels Time-vring current Electromgnetic fiels (or wves) Fr iscovere tht the inuce emf, Vemf (in volts), in n close circuit is equl to the time rte of chnge of the mgnetic flu linkge b the circuit This is clle Fr s w, n it cn be epresse s V emf N 1.1 where N is the number of turns in the circuit n is the flu through ech turn. The negtive sign shows tht the inuce voltge cts in such w s to oppose the flu proucing it. This is known s en s w, n it emphsies the fct tht the irection of current flow in the circuit is such tht the inuce mgnetic file prouce b the inuce current will oppose the originl mgnetic fiel. Fig. 1 A circuit showing emf-proucing fiel Ef n electrosttic fiel Ee

2 TRANFORMER AND MOTIONA EMF Hving consiere the connection between emf n electric fiel, we m emine how Fr's lw links electric n mgnetic fiels. For circuit with single (N = 1), eq. (1.1) becomes V emf In terms of E n B, eq. (1.2) cn be written s N 1.2 Vemf E l B 1.3 where, hs been replce b B n is the surfce re of the circuit boune b the close pth. It is cler from eq. ( 1.3) tht in time-vring sitution, both electric n mgnetic fiels re present n re interrelte. Note tht l n in eq. (1.3) re in ccornce with the right-hn rule s well s tokes's theorem. This shoul be observe in Figure 2. The vrition of flu with time s in eq. (1.1) or eq. (1.3) m be cuse in three ws: 1. B hving sttionr loop in time-vring B fiel 2. B hving time-vring loop re in sttic B fiel 3. B hving time-vring loop re in time-vring B fiel. A. TATIONARY OOP IN TIME-VARYING B FIED (TRANFORMER EMF) This is the cse portre in Figure 2 where sttionr conucting loop is in time vring mgnetic B fiel. Eqution (1.3) becomes B Vemf E l 1.4 t Fig. 2: Inuce emf ue to sttionr loop in time vring B fiel.

3 This emf inuce b the time-vring current (proucing the time -vring B fiel) in sttionr loop is often referre to s trnsformer emf in power nlsis since it is ue to trnsformer ction. B ppling tokes's theorem to the mile term in eq. (1.4), we obtin B E 1.5 t For the two integrls to be equl, their integrns must be equl; tht is, B E 1.6 t This is one of the Mwell's equtions for time-vring fiels. It shows tht the time vring E fiel is not conservtive ( E ). This oes not impl tht the principles of energ conservtion re violte. The work one in tking chrge bout close pth in time-vring electric fiel, for emple, is ue to the energ from the time-vring mgnetic fiel. B. MOVING OOP IN TATIC B FIED (MOTIONA EMF) When conucting loop is moving in sttic B fiel, n emf is inuce in the loop. We recll from eq. (1.7) tht the force on chrge moving with uniform velocit u in mgnetic fiel B is Fm = Qu B 1.7 We efine the motionl electric fiel Em s E m Fm u B 1.8 Q If we consier conucting loop, moving with uniform velocit u s consisting of lrge number of free electrons, the emf inuce in the loop is u B V E l l 1.9 emf m

4 This tpe of emf is clle motionl emf or flu-cutting emf becuse it is ue to motionl ction. It is the kin of emf foun in electricl mchines such s motors, genertors, n lterntors. C. MOVING OOP IN TIME-VARYING FIED This is the generl cse in which moving conucting loop is in time-vring mgnetic fiel. Both trnsformer emf n motionl emf re present. Combining eqution 1.4 n 1.9 gives the totl emf s B V emf E l u B l 1.1 t u B 1.11 E m or from equtions 1.6 n B E u B 1.12 t DIPACEMENT CURRENT For sttic EM fiels, we recll tht H = J 1.13 But the ivergence of the curl of n vector fiel is ienticll ero. Hence,. ( H) = =. J 1.14 The continuit of current requires tht J v t 1.15

5 Thus eqs n 1.15 re obviousl incomptible for time-vring conitions. We must moif eq to gree with eq To o this, we term to eq. 1.13, so tht it becomes H = J + J 1.16 where J is to be etermine n efine. Agin, the ivergence of the curl of n vector is ero. Hence:. ( H) = =. J +. J 1.17 In orer for eq to gree with eq. 1.15, or J J v J t D t D t 1.18 D 1.19 t ubstituting eq into eq results in D H J 1.2 t This is Mwell's eqution (bse on Ampere's circuit lw) for time -vring fiel. The term J = D/t is known s isplcement current ensit n J is the conuction current ensit (J = E) 3. Fig. 3 Two surfces of integrtion showing the nee for J in Ampere s circuit lw The insertion of J into eq ws one of the mjor contribution of Mwell. Without the term J, electromgnetic wve propgtion (rio or TV wves, for emple) woul be impossible. At low frequencies, J is usull neglecte

6 compre with J. however, t rio frequencies, the two terms re comprble. At the time of Mwell, high-frequenc sources were not vilble n eq. 1.2 coul not be verifie eperimentll. Bse on isplcement current ensit, we efine the isplcement current s D I J 1.21 t We must ber in min tht isplcement current is result of time-vring electric fiel. A tpicl emple of such current is tht through cpcitor when n lternting voltge source is pplie to its pltes. PROBEM: A prllel-plte cpcitor with plte re of 5 cm 2 n plte seprtion of 3 mm hs voltge 5 sin 1 3 t V pplie to its pltes. Clculte the isplcement current ssuming = 2. olution: D E V J D t V Hence, I J V V C which is the sme s the conuction current, given b I c Q s D E V V C I cos1 t = cos 1 3 t na

7 EQUATION OF CONTINUITY FOR TIME VARYING FIED Eqution of continuit in point form is. J = -v where, J = conuction current ensit (A/M 2 ) P = volume chrge ensit (C/M 3 ), = vector ifferentil opertor (1/m) v v t Proof: Consier close surfce enclosing chrge Q. There eists n outwr flow of current given b I J This is eqution of continuit in integrl form. From the principle of conservtion of chrge, we hve I J Q From the ivergence theorem, we hve I J J v Thus, J Q

8 B efinition, Q where, = volume chrge ensit (C/m 3 ) o, J t. where. t The volume integrls re equl onl if their integrns re equl. Thus,. J = -. MAXWE' EQUATION FOR TATIC EM FIED Differentil (or Integrl Form Remrks Point) Form. D = v D v v Guss's lw v. B = B B E =- t E l B t s Noneistence of mgnetic monopole Fr s w H = J + H l J D t s Ampere's circuit lw MAXWE EQUATION FOR TIME VARYING FIED These re bsicll four in number. Mwell's equtions in ifferentil form re given b H = D t B E = - t + J

9 .D =.B = Here, H = mgnetic fiel strength (A/m) D = electric flu ensit, (C/m 2 ) (D/t) = isplcement electric current ensit (A/m 2 ) J = conuction current ensit (A/m 2 ) E = electric fiel (V/m) B = mgnetic flu ensit wb/m 2 or Tesl (B/t) = time-erivtive of mgnetic flu ensit (wb/m 2 -sec) B is clle mgnetic current ensit (V/m 2 ) or Tesl/sec P = volume chrge ensit (C/m 3 ) Mwell's equtions for time vring fiels in integrl form re given b H D J E B D B. MEANING OF MAXWE' EQUATION 1. The first Mwell's eqution sttes tht the mgnetomotive force roun close pth is equl to the sum of electric isplcement n, conuction currents through n surfce boune b the pth. 2. The secon lw sttes tht the electromotive force roun close pth is equl to the inflow of mgnetic current through n surfce boune b the pth. 3. The thir lw sttes tht the totl electric isplcement flu pssing through close surfce (Gussin surfce) is equl to the totl chrge insie the surfce. 4. The fourth lw sttes tht the totl mgnetic flu pssing through n close surfce is ero.

10 MAXWE EQUATION FOR TATIC FIED Mwell s Equtions for sttic fiels re: H J H J E E D D B B As the fiels re sttic, ll the fiel terms which hve time erivtives re ero, D tht is, =, t B =. t PROOF OF MAXWE EQUATION 1. From Ampere's circuitl lw, we hve H = J Tke ot prouct on both sies. H =. J As the ivergence of curl of vector is ero, RH =. J = But the eqution of continuit in point form is J t

11 This mens tht if H = J is true, it is resulting in. J =. As the eqution of continuit is more funmentl, Ampere's circuitl lw shoul be moifie. Hence we cn write H = J + F Tke ot prouct on both sies. H =. J +. F tht is,. H = =. J +. F ubstituting the vlue of.j from the eqution of continuit in the bove epression, we get. F + (- ) = or,. F = - The point form of Guss's lw is. D = or,. D = From the bove epressions, we get. F =. D The ivergence of two vectors re equl onl if the vectors re ienticl, tht is, F = D o, Hence prove. H = D + J

12 2. Accoring to Fr's lw, emf = mgnetic flu, (wb) n b efinition, emf E E But B E B t B, B B t Appling toke's theorem to H, we get E E E B Two surfce integrls re equl onl if their integrns re equl, tht is, E = - B Hence prove.

13 3. From Guss's lw in electric fiel, we hve D Q Appling ivergence theorem to H, we get D D Two volume integrls re equl if their integrns re equl, tht is,. D = Hence prove. 4. We hve Guss's lw for mgnetic fiels s B RH is ero s there re no isolte mgnetic chrges n the mgnetic flu lines re close loops. Appling ivergence theorem to H, we get B or,. B = Hence prove. PROBEM 1: Given E = 1 sin (t - ) V/m, in free spce, etermine D, B n H. olution: E = 1 sin (t - ), V/m

14 D = E, = F/m D = 1 sin (t - ), C/m 2 econ Mwell s eqution is E = -B Tht is, E E or, E E E As E = 1 sin (t - ) V/m E Now, E becomes E E = 1 cos (t - ) B t B 1 cos t 1 B sin t or 2, wb / m B 1 n H sint, A/ m

15 PROBEM 2: If the electric fiel strength, E of n electromgnetic wve in free spce is given b E = 2 cos t V/m, fin the mgnetic fiel, H. olution: We hve B/t = - E E E E ) ( E t sin 2 t B sin 2 or, t B cos 2 or, t B H cos 2 12 Thus, t H cos 2 1 m A t H / cos 6 1

16 PROBEM 3: If the electric fiel strength of rio brocst signl t TV receiver is given b E = 5. cos (t - ), V/m, etermine the isplcement current ensit. If the sme fiel eists in meium whose conuctivit is given b (mho)/cm, fin the conuction current ensit. olution: E t TV receiver in free spce = 5. cos (t - ), V/m Electric flu ensit D = E = 5 cos (t - ), V/m The isplcement current ensit J D D t t 5 cos t J = -5 sin (t - ), V/m 2 The conuction current ensit, Jc = E = (mho) /cm = mho /m Jc = cos (t - ) Jc = 1 6 cos (t - ) V/m

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