Chapter7 Time-varying Electromagnetic FieldsDisplacementCurrent,Maxwell'sEquationsBoundaryConditions,PotentialFunctionEnergyFlowDensity,Time-harmonicElectromagneticFieldsComplexVectorExpressionsDisplacementElectricCurrentMaxwell'sEguations3.BoundaryConditionsforTime-varyingElectromagneticFields4.ScalarandVectorPotentials5.SolutionofEguationsforPotentials6.Energy Density and Energy FlowDensity Vector7. Unigueness Theorem8.Time-harmonic Electromagnetic Fields9. Complex Maxwell's Equations1o.ComplexPotentials11.ComplexEnergyDensityandEnergyFlowDensityVectorV
Chapter 7 Time-varying Electromagnetic Fields Displacement Current, Maxwell’s Equations Boundary Conditions, Potential Function Energy Flow Density, Time-harmonic Electromagnetic Fields Complex Vector Expressions 1. Displacement Electric Current 2. Maxwell’s Equations 3. Boundary Conditions for Time-varying Electromagnetic Fields 4. Scalar and Vector Potentials 5. Solution of Equations for Potentials 6. Energy Density and Energy Flow Density Vector 7. Uniqueness Theorem 8. Time-harmonic Electromagnetic Fields 9. Complex Maxwell’s Equations 10. Complex Potentials 11. Complex Energy Density and Energy Flow Density Vector
1.DisplacementElectricCurrentThe displacement current is neither the conduction current nor theconvection current, which are formed by the motion of electric chargesIt is a concept given by J. C. Maxwell.Based on the principle ofelectric charge conservation, we haveopdqfJ ds =-V.J:atatJ.ds =0Forstatic fieldsV.J=0which are called the continuity equations for electric current
1. Displacement Electric Current The displacement current is neither the conduction current nor the convection current, which are formed by the motion of electric charges. It is a concept given by J. C. Maxwell. For static fields d = 0 S J S J = 0 Based on the principle of electric charge conservation, we have t q S = − J dS t = − J which are called the continuity equations for electric current
Fortime-varying electromagnetic fields, since the charges arechanging with time, the electric current continuity principle cannot bederivedfrom staticconsiderations.Nevertheless.an electriccurrentisalways continuous. Hence an extension of earliestconcepts for steadycurrent need to be developedThe currentin a vacuum capacitorisneither the conduction currentnor the11一convectioncurrent.butitisactuallythedisplacementelectriccurrentGauss'law for electrostatic fields, D .ds = q, is still valid for time-varying electricfields, we obtainaDaD:0ds=0atataDObviously, the dimension ofis the same as that of the current densityatV
For time-varying electromagnetic fields, since the charges are changing with time, the electric current continuity principle cannot be derived from static considerations. Nevertheless, an electric current is always continuous. Hence an extension of earliest concepts for steady current need to be developed. Gauss’ law for electrostatic fields, , is still valid for timevarying electric fields, we obtain q S = D dS d 0 = + S S t D J The current in a vacuum capacitor is neither the conduction current nor the convection current, but it is actually the displacement electric current. Obviously, the dimension of is the same as that of the current density. t D = 0 + t D J
British scientist, James Clerk Maxwell named D the density of theatdisplacement current, denoted as J., so thataDatWe obtain(J+J.)-ds =0V(J+J)=0The introduction of the displacement current makes the timevarying total current continuous, and the above equations are calledthe principle oftotal current continuityThe density of the displacement current is the time rate of changeoftheelectricflux density,henceFor electrostatic fields,D- O, and the displacement current is zero.atIn time-varying electric fields, the displacement current is larger ifthe electric field is changing more rapidlyIn imperfect dielectrics,J, >> J., while in a good conductor, J, << J.√
t = D Jd ( ) d 0 + d = S We obtain J J S (J + Jd ) = 0 British scientist, James Clerk Maxwell named the density of the displacement current, denoted as Jd , so that t D The introduction of the displacement current makes the timevarying total current continuous, and the above equations are called the principle of total current continuity. The density of the displacement current is the time rate of change of the electric flux density, hence For electrostatic fields, = 0 , and the displacement current is zero. t D In time-varying electric fields, the displacement current is larger if the electric field is changing more rapidly. In imperfect dielectrics, , while in a good conductor, . Jd Jc d c J J
Maxwell considered thatthe displacementcurrent mustalsoproducemagnetic fields,andit should be includedin the Ampere circuitallaw, sothatf H .dl =[,(J +Ja) dsaDaDfH d/ -J,(J + oDi.e.)·dsVxH=JatatWhich are Ampere's circuitallaw with the displacement current.Itshows that a time-varying magnetic field is produced by the conductioncurrent, the convectioncurrent,and the displacement currentThe displacementcurrent, which results from time-varying electricfield, produces a time-varying magnetic field.The law of electromagneticinduction shows that a time-varyingmagnetic field can produce a time-varying electric field.Maxwell deduced the coexistence of a time-varying electric field anda time-varying magnetic field, and they resultin an electromagneticwavein space.This prediction was demonstratedin 1888byHertzK
Maxwell considered that the displacement current must also produce magnetic fields, and it should be included in the Ampere circuital law, so that H dl (J J ) dS d = + l S S D H dl (J ) d = + l S t t = + D i.e. H J Which are Ampere’s circuital law with the displacement current. It shows that a time-varying magnetic field is produced by the conduction current, the convection current, and the displacement current. The displacement current, which results from time-varying electric field, produces a time-varying magnetic field. Maxwell deduced the coexistence of a time-varying electric field and a time-varying magnetic field, and they result in an electromagnetic wave in space. This prediction was demonstrated in 1888 by Hertz. The law of electromagnetic induction shows that a time-varying magnetic field can produce a time-varying electric field
2.Maxwell'sEquationsFor the time-varying electromagnetic field, Maxwell summarizedthefollowingfourequations:ThedifferentialformThe integralformaDaDVxH=J.fH .dl -[,(J).dsatataBaBVXEfE·d --dsatISatV.B=0fB.ds=0V.D=pf D.ds =q
2. Maxwell’s Equations For the time-varying electromagnetic field, Maxwell summarized the following four equations: S D H dl (J )d = + l S t S B E dl d = − l S t d = 0 S B S q S = D dS The integral form t = + D H J t = − B E B = 0 D = The differential form
aDV.B=0VxH=JataBVxE:V.D=patThe time-varying electric field is both divergent and curly, and thetime-varying magnetic field is solenoidal and curly.Nevertheless, thetime-varying electric field and the time-varying magnetic field cannotbe separated, and the time-varying electromagnetic field is divergentand curlyIn a source-free region, the time-varying electromagnetic field issolenoidal.The electric field lines and the magnetic field lines are linked witheach other, forming closed loops, and resultingin an electromagneticwaveinspaceThe time-varying electric field and the time-varying magnetic fieldareperpendicularto each other7
The time-varying electric field is both divergent and curly, and the time-varying magnetic field is solenoidal and curly. Nevertheless, the time-varying electric field and the time-varying magnetic field cannot be separated, and the time-varying electromagnetic field is divergent and curly. In a source-free region, the time-varying electromagnetic field is solenoidal. The electric field lines and the magnetic field lines are linked with each other, forming closed loops, and resulting in an electromagnetic wave in space. The time-varying electric field and the time-varying magnetic field are perpendicularto each other. t = + D H J t = − B E B = 0 D =
In order to describe more completely the behavior of time-varyingelectromagnetic fields, Maxwell's eguations need to be supplementedby the charge conservation equation and the constitutiverelationsapV.JD=εEB=uHJ=oE+Jatwhere J' stands for the impressed source producing the time-varyingelectromagneticfieldThe four Maxwell's equations are not independent.Equations 4 and3 can be derived from Equation 1 and 2, respectively,and viseversa.Forstaticfields,wehaveaEaDaHaB=0atatatatMaxwell's eguationsbecometheformerequationsforelectrostaticfield and steady magnetic field. Furthermore, the electric field and themagneticfieldareindependenteachother
In order to describe more completely the behavior of time-varying electromagnetic fields, Maxwell’s equations need to be supplemented by the charge conservation equationand the constitutive relations: t = − J D = E B = H J = E + J The four Maxwell’s equations are not independent. Equations 4 and 3 can be derived from Equation 1 and 2, respectively, and vise versa. For static fields, we have = 0 = = = t t t t E D H B Maxwell’s equations become the former equationsfor electrostatic field and steady magnetic field. Furthermore, the electric field and the magnetic field are independent each other. where stands for the impressed source producing the time-varying electromagnetic field. J
As the founder of relativity, Albert Einstein (1879-1955), pointed outin his book“TheEvolutionofPhysics"thatThe formulation of these equations is the most important event inphysics since Newton's time, and they are the quantitative mathematicaldescription of the laws of the field. Their content is much richer than wehave been able to indicate, and the simple form conceals a depthrevealedonlybycareful study""These equations are the laws representing the structure of the field.They do not, as in Newton's laws, connect two widely separated events:they do not connect the happenings here with the conditions there"The field here and now depends on the field in the immediateneighborhood at a time just past. The equations allow us to predictwhat will happen a little further in space and a little later in time, if weknowwhathappens hereandnow
“These equations are the laws representing the structure of the field. They do not, as in Newton’s laws, connect two widely separated events; they do not connect the happenings here with the conditionsthere”. As the founder of relativity, Albert Einstein (1879-1955), pointed out in his book “The Evolutionof Physics” that “The field here and now depends on the field in the immediate neighborhood at a time just past. The equations allow us to predict what will happen a little further in space and a little later in time, if we know what happens here and now” “The formulation of these equations is the most important event in physics since Newton’s time, and they are the quantitative mathematical description of the laws of the field. Their content is much richer than we have been able to indicate, and the simple form conceals a depth revealed only by carefulstudy
Maxwell's eguationshave madeimportantimpact onthehistoryof mankind, besidestheadvancementof scienceandtechnology.As American physicist,RichardP.Feynman,saidin his bookThe Feynman Lectures on Physics",that"From alongview of thehistory of mankind--seen from say, ten thousand years from now-there can be littledoubt that the most significantevent of the19.th centurywill be judged as Maxwell's discoveryof the laws ofelectrodynamics. The American civil war will pale into provincialinsignificance in comparison with this important scientific event ofthe samedecade
Maxwell’s equations have made important impact on the history of mankind, besides the advancement of science and technology. As American physicist, Richard P. Feynman, said in his book “The Feynman Lectures on Physics”, that “From along view of the history of mankind──seen from say, ten thousand years from now ──there can be little doubt that the most significant event of the 19- th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American civil war will pale into provincial insignificance in comparison with this important scientific event of the same decade