TIME VARYING MAGNETIC FIELDS AND MAXWELL’S EQUATIONS
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1TIME VARYING MAGNETIC FIELDS AND MAXWELL’S EQUATIONS Hello!! Welcome to E-learningSubject:TIME VARYING MAGNETIC FIELDS AND MAXWELL’S EQUATIONSPresented by:DR. H. N. SURESHAssistant ProfessorDepartment of E&E EngineeringMalnad College of EngineeringHassan (O) (R)
2TIME VARYING MAGNETIC FIELDS INTRODUCTIONElectrostatic fields are usually produced by static electric charges.Magnetostatic fields are due to motion of electric charges with uniform velocity (direct current) or static magnetic charges (magnetic poles).Time-varying fields or waves are usually due to accelerated charges or time-varying current.
3Stationary charges Electrostatic fields Steady current Magnetostatic fieldsTime-varying current Electromagnetic fields(or waves)FARADAY’S DISCOVERY :Induced emf, Vemf, in any closed circuit is equal to the time rate of change of the magnetic flux linkage in the circuit
4Faraday’s Law is expressed as 1.1where, N is the number of turns in the circuit is the flux through each turnNegative sign shows that the induced voltage acts in such a way as to oppose the flux producing it.- LENZ’S LAW- Direction of current flow in the circuit is such that theinduced magnetic filed produced by the inducedcurrent will oppose the original magnetic field.
5TRANSFORMER AND MOTIONAL EMFS . Connection between emf and electric field (Previous Section), . Link between electric and magnetic fields throughFaraday's law (To be explored now)For a circuit with single conductor (N = 1)1.2In terms of E and B, equation can be written as1.3
6where, has been replaced by integral B where, has been replaced by integral B.ds and S is the surface area of the circuit bounded by the closed path L.It is clear from eq. (1.3) that in a time-varying situation, both electric and magnetic fields are present and are interrelated.The variation of flux with time as in eq. (1.1) or eq. (1.3) may be caused in three ways:
72. By having a time-varying loop area in a static B field 1. By having a stationary loop in a time-varying B field2. By having a time-varying loop area in a static B field3. By having a time-varying loop area in a time-varying B field.A. STATIONARY LOOP IN TIME-VARYING B FIELD(TRANSFORMER EMF)This is the case portrayed in Figure 1, where a stationary conducting loop is in a time varying B field.Equation (1.3) becomes1.4Fig. 1: Induced emf due to a stationaryloop in a time varying B field.
8Transformer emf (It is due to transformer action) EMF induced by the time-varying current (producing the time-varying B field) in a stationary loop is -Transformer emf (It is due to transformer action)Applying Stoke’s theorem to the middle term in eq. (1.4), one can obtain1.5For the two integrals to be equal, their integrands must be equal; i.e.,1.6
9This is one of the Maxwell's equations for time- varying fields.It shows that the time varying E field is notconservative ( x E 0).This does not imply that the principles of energyconservation are violated.The work done in taking a charge about a closedpath in a time-varying electric field, for example, isdue to the energy from the time-varying magneticfield.
10B. MOVING LOOP IN STATIC B FIELD (MOTIONAL EMF) When a conducting loop is moving in a static B field,an emf is induced in the loop.Force on a charge moving with uniform velocity u in amagnetic field B isFm = Qu x BMotional electric field Em is defined as1.8
11Considering a conducting loop, moving with uniform velocity u as consisting of a large number of free electrons, the emf induced in the loop is1.9This type of emf is calledMotional emf or flux-cutting emf(It is due to motional action)It is the kind of emf found in electrical machines such asMotors, Generators and Alternators.
12C. MOVING LOOP IN TIME-VARYING FIELD This is the general case in which a moving conductingloop is in a time-varying magnetic field.Both transformer emf and motional emf are present.Combining equation 1.4 and 1.9 gives the total emf as1.101.12
13DISPLACEMENT CURRENT 1.11 or from eqs. 1.6 and 1.11. 1.12 For static EM fields, one can recall that x H = J
14But the divergence of the curl of any vector field is identically zero.Hence, . ( x H) = 0 = . JThe continuity of current requires that1.15Thus, equations 1.14 and 1.15 are obviouslyincompatible for time-varying conditions.We must modify equation 1.13 to agree with equationTo do this, add a term to equation 1.13, so that it becomes
15 x H = J + Jd 1.16Where, Jd is to be determined and defined. Again, the divergence of the curl of any vector is zero. Hence: . ( x H) = 0 = . J + . JdFor equation 1.17 to agree with equation 1.15,1.18
16or1.19Substituting eq into eq results in1.20This is Maxwell's equation for a time-varying field.(based on Ampere's circuit law)where,Jd = D/t is known as displacement current densityand J is the conduction current density
17THANK YOU DR. H. N. SURESH For any clarification: Assistant Professor Department of E&E EngineeringMalnad College of EngineeringHassan (O) (R) (M)