TIME VARYING MAGNETIC FIELDS AND MAXWELL’S …

1 TIME VARYING MAGNETIC FIELDS ANDMAXWELL S EQUATIONSI ntroductionElectrostatic FIELDS are usually produced by static electric charges whereasmagnetostatic FIELDS are due to motion of electric charges with uniform velocity(direct current) or static MAGNETIC charges ( MAGNETIC poles); time- VARYING fieldsor waves are usually due to accelerated charges or time- VARYING current. Stationary charges Electrostatic FIELDS Steady current Magnetostatic FIELDS Time- VARYING current Electromagnetic FIELDS (orwaves)Faraday discovered that the induced emf, Vemf(in volts), in any closed circuit isequal to the time rate of change of the MAGNETIC flux linkage by the circuitThis is called Faraday s Law, and it can be expressed asdtdNdtdVemf N is the number of turns in the circuit and is the flux through eachturn. The negative sign shows that the induced voltage acts in such a way as tooppose the flux producing is known as Lenz s Law, and it emphasizesthe fact that thedirection of current flow in the circuit is such that the inducedmagnetic filed producedby the induced current will oppose the originalmagnetic circuit showing emf-producing field Efand electrostatic field EeTRANSFORMER AND MOTIONAL EMFSH aving considered the connection between emf and electric field , we mayexamine how Faraday's law links electric and MAGNETIC FIELDS .

2 For a circuitwith a single (N = 1), eq. ( ) becomesdtdNVemf terms ofEandB,eq.( ) can be written as , has been replaced by SdSBand S is the surface area of the circuitbounded by the closed path L. It is clear from eq. ( ) that in a time-varyingsituation, both electricand MAGNETIC FIELDS are present and are that dland dSin eq. ( ) are in accordance with the right-hand rule aswell as Stokes's theorem. This should be observed in Figure2. The variation offlux with time as in eq. ( ) or eq. ( ) may be caused in three ways:1. By having a stationary loop in a time-varyingBfield2. By having a time- VARYING loop area in a having a time- VARYING loop area in a STATIONARY LOOPIN TIME- VARYING B field (TRANSFORMER EMF)This is the case portrayed in Figure2where a stationary conducting loop is ina time VARYING magneticBfield. Equation ( ) becomes :Induced emf due to a stationary loop in a time emf induced by the time- VARYING current (producing the time-varyingBfield) in a stationary loop is often referred to astransformer emfin poweranalysis since it is due to transformer action.

3 By applying Stokes's theorem tothe middle term in eq. ( ), we obtain the two integrals to be equal, their integrands must be equal; that is,tBE is one of the Maxwell's equations for time- VARYING FIELDS . It shows that thetimevarying E field is not conservative ( x E 0). This does not imply that theprinciples of energy conservation are violated. The work done in taking acharge about a closed path in a time- VARYING electric field , for example, is dueto the energy from the time- VARYING MAGNETIC MOVING LOOP IN STATIC B field (MOTIONAL EMF)When a conducting loop is moving in a staticBfield, an emf is induced in theloop. We recall from eq. ( ) that the force on a charge moving with uniformvelocityuin a MAGNETIC fieldBisFm= define themotional electricfieldEmasBuQFEmm we consider a conducting loop, moving with uniform velocityuas consistingof a large number of free electrons, the emf induced in the loop is type of emf is calledmotional emf orflux-cutting emfbecause it is due tomotional action.

4 It is the kind of emf found in electrical machines such asmotors, generators, and MOVING LOOP IN TIME- VARYING FIELDT hisis the general case in which a moving conducting loop is in a time-varyingmagnetic field . Both transformer emf and motional emf are present. and the total emf as BuEm from and ButBE CURRENTFor static EM FIELDS , we recall that xH= the divergence of the curl of any vector field is identically , . ( x H) = 0 = . continuity of current requires that0 tJv obviously incompatible for must modify agree with To do this, we adda term to ,so that it becomes x H = J + Jdis to be determined and defined. Again, the divergence of the curl ofany vector is zero. Hence: . ( x H) = 0 = . J + . order for agree with , tDDttJJvd intDJH is Maxwell's equation (based on Ampere's circuit law) for a Jd= D/ tis known asdisplacement current density andJis theconduction currentdensity (J = E) surfaces of integration showing the need for Jdin Ampere s circuit lawThe insertion of Jdinto one of the major contribution of the term Jd, electromagnetic wave propagation (radio or TV waves, forexample) would be impossible.

5 At low frequencies, Jdis usually neglectedcompared with J. however, at radio frequencies, the two terms are the time of Maxwell, high-frequency sources were not available and not be verified on displacement current density, we define the displacement current as must bear in mind that displacement current is a result of time-varyingelectric field . A typical example of such currentis thatthrough a capacitorwhen an alternating voltage source is applied to its :A parallel-plate capacitor with plate area of 5 cm2and plateseparation of 3 mm has a voltage 50 sin 103tV applied to its plates. Calculatethe displacement current assuming = 2 :dVED dtdVdtDJd Hence,dtdVCdtdVdSSJIdd which is the same as the conduction current, given bydtdVCdtdVdSdtdESdtdDSdtdSdtdQIsc tId3334910cos501010310536102 = cos 103tnAEQUATION OF CONTINUITY FOR TIME VARYING FIELDSE quation of continuity in pointform is.

6 J =- vwhere,J =conduction current density (A/M2)P = volume charge density (C/M3),tvv = vector differential operator (1/m) zayaxazyx Proof:Consider a closed surface enclosing a exists an outwardflow of current given by SdSJIThis is equation of continuity inintegral the principle of conservation of charge, we havedtdQdSJIS From the divergence theorem, wehave vSdJdSJI Thus, dtdQdJ By definition, dQwhere, = volume charge density (C/m3)So, .The volume integrals are equal only if theirintegrands are , . J =-. MAXWELL'S EQUATIONSFOR STATIC EM FIELDSD ifferential (orPoint) FormIntegral FormRemarks . D = v SvvdvdSD Gauss's law . B = 0 SdSB0 Nonexistence of magneticmonopole x E =-tB LsdSBtdlEFaraday s Law x H = J +tD LsdSJdlHAmpere's circuit lawMAXWELL S EQUATIONS FOR TIME VARYING FIELDST hese are basically four in 's equations indifferential formare given by xH =tD + J xE =-tB .D = .B = 0 Here,H = MAGNETIC field strength (A/m)D = electric flux density, (C/m2)( D/ t)= displacement electric current density (A/m2)J =conduction current density (A/m2)E =electric field (V/m)B = MAGNETIC flux density wb/m2or Tesla( B/ t)=time-derivative of MAGNETIC flux density (wb/m2-sec)B is called MAGNETIC current density (V/m2) or Tesla/secP =volume charge density (C/m3)Maxwell's equations for time VARYING FIELDS inintegral formare given by LSdSJDdLH.

7 LSdSBdLE SddSD SdSB0 MEANING OF MAXWELL'S first Maxwell's equation states that the magnetomotive force arounda closed path is equal to the sum of electric displacement and,conduction currents through any surface bounded by the second law states that the electromotive force around a closed pathis equal to the inflow of MAGNETIC current through any surface boundedby the third law states that the total electric displacement flux passingthrough a closed surface (Gaussian surface) is equal to the total chargeinside the fourth law states that the total MAGNETIC flux passing through anyclosed surface is S EQUATIONS FOR STATIC FIELDSM axwell s Equations for static FIELDS are: SLdSJdLHJH00 LdLEE ddSDDS00 SdSBBAs the FIELDS are static, all the field terms which have time derivatives are zero,that is,tD = 0,tB = OF MAXWELLS Ampere's circuital law, we have xH =JTake dotproduct on both sides.

8 XH = .JAs the divergence of curl of a vector is zero,RHS = .J= 0 But the equation of continuity in point form is tJThis means that if x H =Jis true, it is resulting in .J = the equation of continuity is more fundamental, Ampere's circuital lawshould be modified. Hence we can write xH =J+FTake dot product on both sides . xH = .J+ .Fthat is, . xH =0= .J+ .FSubstituting the value of .Jfrom the equation of continuity in the aboveexpression, we get . F + (- ) = 0or, . F =- The point form of Gauss's law is .D = or, . D = From the above expressions, we get .F = .DThe divergence of two vectors are equal only if the vectors are identical,that is,F = DSo, x H =D + JHence According to Faraday's law,dtdemf = MAGNETIC flux, (wb)and by definition, LdLEemfdtddLEL But SdSB SLdStBdLE StBBdSB,Applying Stoke's theorem to LHS, we get SLdSEdLE SSdSBdSETwo surface integrals are equal only if their integrands are equal,that is, xE =-BHence From Gauss's law inelectric field , we have dQdSDSA pplying divergence theorem to LHS, we get ddDdSDSTwo volume integrals are equal if their integrands are equal,that is.

9 D = Hence have Gauss's law for MAGNETIC FIELDS as SdSB0 RHS is zero as there are no isolated MAGNETIC charges and the MAGNETIC fluxlines are closed divergence theorem to LHS, we get 0dBor, .B =0 Hence 1:GivenE = 10 sin ( t- y) ayV/m,in free space, determine D, B :E = 10 sin ( t- y) ay,V/mD = 0E, 0= x 10-12F/mD = 10 0sin ( t- y) ay, C/m2 Second Maxwell s equation is x E =-BThat is,00yzyxEzyxaaaE or, yzyxExaEzaE0 AsEy= 10 sin ( t- z) V/m0 xEyNow, x EbecomesxyazEE = 10 cos ( t- z) axtB xadtztB cos10or 2/,sin10mwbaztBz and mAaztBHz/,sin1000 PROBLEM 2:If the electric field strength, E of an electromagnetic wave in freespace is given by E = 2 cos yazt 0 V/m,find the MAGNETIC field , :We have B/ t =- x E00yzyxEzyxaaa yzyyxExaaEza)0(xyazE xazt 00sin2 xadtztB00sin2 or,xaztB 00cos2 or,xaztBH 0000cos2 120000 Thus,xaztH 00cos2 0001 mAaztHx/cos6010 PROBLEM 3:If the electric field strength of a radio broadcast signal at a TVreceiver is given byE = cos ( t- y) az, V/m,determine the displacement current density.

10 If the same field exists in amedium whose conductivity is given by x 103(mho)/cm, find theconduction current :E at a TVreceiver in free space= cos ( t- y) az, V/mElectric flux densityD = 0E = 5 0cos ( t- y) az, V/mThe displacement current densitytDDJd zaytt cos50Jd=-5 0 sin ( t- y) az, V/m2 The conduction current density,Jc= E = x 103(mho) /cm= 2 x 105mho /mJc= 2 x 105x 5 cos ( t- y) azJc= 106cos ( t- y) azV/m2.