Maxwell’s Equations & The Electromagnetic Wave Equation.

1 SMR 1826-2 Preparatory School Winter College on Fibre Optics, Fibre Lasers and Sensors maxwell s Equations &The Electromagnetic Wave EquationImrana Ashraf Zahid 5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors1 maxwell s Equations &The Electromagnetic Wave EquationDr. Imrana Ashraf ZahidQuaid-i-Azam University, IslamabadPakistan5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors2 maxwell s Equations Introduction Historical background Electrodynamics before maxwell maxwell s correction to Ampere s law General form of maxwell s Equations maxwell s Equations in vacuum maxwell s Equations inside matter5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors3 Introduction In electrodynamics maxwell s Equations are a set of four Equations , that describes the behavior of both the electric and magnetic fields as well as their interaction with matter maxwell s four Equations express How electric charges produce electric field (Gauss s law) The absence of magnetic monopoles How currents and changing electric fields produces magnetic fields (Ampere s law)

2 How changing magnetic fields produces electric fields(Faraday s law of induction)5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors4 Historical Background 1864 maxwell in his paper A Dynamical Theory of the Electromagnetic Field collected all four Equations 1884 Oliver Heaviside and Willard Gibbs gave the modern mathematical formulation using vector calculus. The change to vector notation produced a symmetric mathematical representation, that reinforced the perception of physical symmetries between the various School on Fiber Optics, Fiber Lasers and Sensors5 Nomenclature E= Electric field D= Electric displacement B= Magnetic flux density H= Auxiliary field U= Charge density j= Current density P0(permeability of free space) = 4Su10-7 H0(permittivity of free space) = c (speed of light) = School on Fiber Optics, Fiber Lasers and Sensors6 Electrodynamics Before MaxwellJBivtBEiiiBiiEiooPHU u ww u x x )()(0)()(ABtAVEu ww Gauss s LawNo nameFaraday s LawAmpere s Law5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors7 Electrodynamics Before maxwell (Cont d)Apply divergence to (iii).

3 ,issidehandleftThe x x ww ww x u x BBttBEApply divergence to (iv) JBox u x P5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors8 The left hand side is zero, because divergence of a curl is zero. The right hand side is zero for steady currents , In electrodynamics from conservation of chargeElectrodynamics Before maxwell (Cont d)0 x J0 ww ww x ttJUUis constant at any point in space which is School on Fiber Optics, Fiber Lasers and Sensors9 maxwell s Correction to Ampere s LawConsider Gauss s Law tEttEtEooowwx ww ww x ww x HUUHUHtEtDoww wwHDisplacement currentThis result along with Ampere s law and the conservation of charge equation suggest that there are actually two sources of magnetic current density and displacement School on Fiber Optics, Fiber Lasers and Sensors10 maxwell s Correction to Ampere s Law (Cont d)Amperes law with maxwell s correctiontEJBoooww u HPP5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors11 General Form of maxwell s EquationsDifferential FormIntegral Form x xx x x xSooencoCSCSVoSSdEdtdIldBSdBdtdldESdBdVS dEHPPUH&010tEJBtBEBE ooooww u ww u x x HPPHU5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors12 maxwell s Equations in vacuumtEBtBEBE ooww u ww u x x HP00 The vacuum is a linear, homogeneous, isotropic and dispersion less medium Since there is no current or electric charge is present in the vacuum, hence maxwell s Equations reads as These Equations have a simple solution interms of traveling sinusoidal waves , with the electric andmagnetic fields direction orthogonal to each other and the direction of travel 5-02-2007 Preparatory School on Fiber Optics.

4 Fiber Lasers and Sensors13 maxwell s Equations Inside MatterMaxwell s Equations are modified for polarized and magnetized linear materials the polarization Pand magnetization Mis given by HMEmeoFFH And the DandBfields are related toEandHby .andmaterialoflitysusceptibimagneticthei smaterial,oflitysusceptibielectrictheisW here11meomooeoHHMHBEEPEDFFPPFPHHFH& 5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors14 maxwell s Equations Inside Matter (Cont d) For polarized materials we have bound charges in addition to free chargesPnPbbx x UV MJnMKbbu u For magnetized materials we have bound currents5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors15 maxwell s Equations Inside Matter (Cont d) In electrodynamics any change in the electric polarization involves a flow of bound charges resulting in polarization current JPPolarization current density is dueto linear motion of charge when the Electric polarization changes tPJpww pbfbftJJJJ tdensitycurrentTotaldensitychargeTotalUU U5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors16 maxwell s Equations Inside Matter (Cont d) maxwell s Equations inside matter are written astEJJJBtBEBE oobopofootww u ww u x x HPPPPHU0 DtJHPEtJMBtEMtPJBfofoofoww u ww u ww u ww u HPHP5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors17 maxwell s Equations Inside Matter (Cont d)

5 In non-dispersive, isotropic media and are time-independent scalars, and maxwell s Equations reduces totEJHtHEHEww u ww u x x HPPUH05-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors18 maxwell s Equations Inside Matter (Cont d) In uniform (homogeneous) medium and areindependent of position, hence maxwell s Equations reads asGenerally, and can be rank-2 tensor (3X3 matrices) x xww u x xww u x x x x SencffSencffSdDdtdIldHtEJHSdHdtdldEtHESd HHQSdDDHPPU5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors19 The Electromagnetic wave equation (EM Wave) The EM wave from maxwell s Equation Solution of EM wave in vacuum EM plane wave Polarization energy and momentum of EM wave Inhomogeneous wave equation5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors20 The Electromagnetic Wave from maxwell s EquationsTake curl of][tBEtBEww u u u ww u Change the order of differentiation on the ][BtEu ww u u 5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors21 The Electromagnetic Wave from maxwell s Equations (cont d)AstEBooww u HPSubstituting for we haveBu 22][][][][tEEtEtEtBEooooww u u wwww u u ww u u HPHP Assuming that oand oare constant in time5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors22 The Electromagnetic Wave from maxwell s Equations (cont d)22tEEww u u 222)

6 (tEEEooww x HP0 x EUsing the vector identitybecomes,In free spaceAnd we are left with the wave equation 0222 ww tEEooHP5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors23 The Electromagnetic Wave from maxwell s Equations (cont d)Similarly the wave equation for magnetic field0222 ww tBBoo&&HPoocHP1 where,5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors24 Solution of Electromagnetic waves in Vacuum0222 ww tBBoo&&HPThe solutions to the wave Equations , where there is nosource charge is present0222 ww tEEoo&&HPcan be plane waves , obtained by method of separation of variables5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors25 Solution of Electromagnetic waves in Vacuum (Cont d) tiotioeBBeEEZZ rkrk&&&&&&&&Where Eoand Boare the complex amplitudes of electric and magnetic fields and related to each other by relation) (1ooEkcBu Where is a propagation 5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors26 Electromagnetic Plane waves Plane Electromagnetic waves can be expressed as EcneEcBneEEtiotiou u kkrkrk 1 1 ZZ&&&&&&&&Where is the polarization 5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors27 Electromagnetic Plane wavesThe real electric and magnetic fields in a monochromatic plane wave with propagation vectork and polarization n are therefore nktrkEctrBntrkEtrEoo )cos(1, )cos(,u x x ZZ5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors28 Polarization The polarization is specified by the orientation of the Electromagnetic field.)

7 The plane containing the electric field is called the plane of School on Fiber Optics, Fiber Lasers and Sensors29 Polarization (Cont d) Can be horizontal, vertical, circular, or ellipticalxyzEHorizontal PolarizationElectric FieldMagnetic FieldElectromagneticWavexyzEVertical Polarization5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors30 Polarization (Cont d)5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors31 energy and Momentum of Electromagnetic WavesThe energy per unit volume stored in Electromagnetic field is 22121 BEUooPHIn the case of monochromatic plane wave)(cos12222222tkxEEUEEcBoooooZHHHP 5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors32 energy and Momentum of Electromagnetic waves (Cont d) As the wave propagates, it carries this energy along with it. The energy flux density ( energy per unit area per unit time) transported by the field is given by the poynting vector BESou P1 For monochromatic plane waves icUitkxEcSoo cos22 ZH5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors33 Homogenous Wave Equations Inside MatterMatterVacuum2222222222221111tBBtBB tEEtEEooooww ww ww ww PHHPPHHP5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors34 Homogenous Wave Equations Inside Matter ( )rrrrvHPHPHHPPPH11110000 ncv =c= nPermittivity: = r o( ris dielectric constant)Permeability.

8 = r o( ris relative permeability 1n=Refractive Index5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors35 Inhomogeneous Electromagnetic Wave EquationInside linear dielectric medium with no free charge present, maxwell s Equations reads as(iv)(iii)(ii)0(i)0tDHtBEBDww u ww u x x Where,HBPEDooPH and5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors36 Inhomogeneous Electromagnetic wave equation (Cont d)Taking curl of (iii) ][][][HtBtEowwu wwu u u PUsing (iv)22222222222222222222221111][tPctEcEt PctEcEtPtEEtDEEooooooww ww ww ww ww ww ww x HHPHPPS ource term5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors37 Solution of Inhomogeneous Electromagnetic Wave Equation222222211tPctEcEoww ww HInhomogeneous wave equation can be solved with the help of Green s Theorem5-02-2007 Preparatory School on Fiber Optics, Fiber Lasers and Sensors38 THANK YOU)