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