Preparatory School to the Winter College on Optics in Environmental Science January 2009

Transcription

1 017-1 Preparatry Schl t the Winter Science 6-30 January 009 Review f lectrdynamics: Frm Maxwell's quatins t the lectrmagnetic waves Ashraf Zahid I. Quaid-I-Azam University Pakistan

2 RVIW OF LCTODYNAMICS: FROM MAXWLL S QUATIONS TO TH LCTROMAGNTIC WAVS Dr. Imrana Ashraf Zahid Quaid-i-Azam University, Islamabad Pakistan Preparatry Schl t the Winter 1

3 Layut lectrstatic : Revisited Magnet- static : Revisited Intrductin t Maxwell s equatins lectrdynamics befre Maxwell Maxwell s crrectin t Ampere s law General frm f Maxwell s equatins Maxwell s equatins in vacuum Maxwell s equatins inside matter The lectrmagnetic wave Plarizatin nergy and Mmentum f lectrmagnetic Waves Preparatry Schl t the Winter

4 Nmenclature lectric field D lectric displacement B Magnetic flux density H Auxiliary field ρ Charge density j Current density μ 0 (permeability f free space) 4π 10-7 T-m/A ε 0 (permittivity f free space) N-m / C c (speed f light) m/s Preparatry Schl t the Winter 3

5 lectrstatics Intrductin lectrstatic field : Statinary charges prduce electric fields that are cnstant in time. The thery f static charges is called electrstatics. Statinary charges Cnstant lectric field; Preparatry Schl t the Winter 4

6 lectrstatic :Revisited Culmbs Law F 4 1 πε 0 ε qq r N C rˆ m q Surce Charge r Permittivity f free space Q Test Charge Preparatry Schl t the Winter 5

7 The lectric Field F Q 1 ( P) 4πε 0 n i 1 q r ˆ i r i i y r i q i q q n r ri P Field Pint - the electric field f the surce charges. x Physically (P) Is frce per unit charge exerted n a test charge placed at P. z Preparatry Schl t the Winter 6

8 The lectric Field: cnt d r P ( P) 1 rˆ r 4πε0 Line λdl λ is the line charge density r P ( P) 1 rˆ r 4πε 0 Surface σda σ is the surface charge density Preparatry Schl t the Winter 7

9 The lectric Field: cnt d r P ( P) 1 4πε rˆ r 0 Vlume ρdτ ρ is the vlume charge density Preparatry Schl t the Winter 8

10 lectric Ptential The wrk dne in mving a test charge Q in an electric field frm pint P 1 t P with a cnstant speed. W W Frce dis tan ce P Q dl negative sign - wrk dne is against the field. Fr any distributin f fixed charges. p 1 dl 0 The electrstatic field is cnservative Preparatry Schl t the Winter 9

11 lectric Ptential: cnt d Stkes s Therem gives 0 V where V is Scalar Ptential The wrk dne in mving a charge Q frm infinity t a pint P where ptential is V W QV V Wrk per unit charge Vlts jules/culmb Preparatry Schl t the Winter 10

12 lectric Ptential : cnt d Field due t a single pint charge q at rigin V qdr q 4πε r πε r r F r 1 r 1 V r Preparatry Schl t the Winter 11

13 Gauss s Law da 1 ε 0 Qenc Differential frm f Gauss s Law ε 0 ρ Pissn s quatin V ρ ε 0 Laplace's quatin V 0 Preparatry Schl t the Winter 1

14 lectrstatic Fields in Matter Matter: Slids, liquids, gases, metal, wd and glasses - behave differently in electric field. Tw Large Classes f Matter (i) Cnductrs (ii) Dielectric Cnductrs: Unlimited supply f free charges. Dielectrics: Charges are attached t specific atms r mlecules- N free charges. Only pssible mtin - minute displacement f psitive and negative charges in ppsite directin. Large fields- pull the atm apart cmpletely (inizing it). Preparatry Schl t the Winter 13

15 Plarizatin A dielectric with charge displacements r induced diple mment is said t be plarized Induced Diple Mment p α The cnstant f prprtinality α is called the atmic plarizability P diple mment per unit vlume Preparatry Schl t the Winter 14

16 The Field f a Plarized Object Ptential f single diple p is V dv V 1 rˆ p πε 4 0 r 1 P rˆ dτ 4πε r 0 vlume P da 4 πε 0 r r surface vlume Ptential due t diples in the dielectric r ( P ) d τ Preparatry Schl t the Winter 15

17 The Field f a Plarized Object: cnt d σ V b ρ b P nˆ P 1 1 σ b da 4 πε 0 r surface Bund charges at surface Bund charges in vlume vlume 1 r ρ b d τ The ttal field is field due t bund charges plus due t free charges Preparatry Schl t the Winter 16

18 Gauss s law in Dielectric ffect f plarizatin is t prduce accumulatins f bund charges. The ttal charge density ρ ρ f + ρ b D da Q fenc Frm Gauss s law ε ρ ρ + 0 D ρ f b ρ f Q fenc -Free charges enclsed Displacement vectr D ε 0 + P Preparatry Schl t the Winter 17

19 Magnetstatics : Revisited Magnetstatics Steady current prduce magnetic fields that are cnstant in time. The thery f cnstant current is called magnetstatics. Steady currents Cnstant Magnetic field; Preparatry Schl t the Winter 18

20 Magnetic Frces Lrentz Frce F q [ + ( v B) ] The magnetic frce n a segment f current carrying wire is F mag F mag ( ) I B dl I B ( ) dl Preparatry Schl t the Winter 19

21 quatin f Cntinuity The current crssing a surface s can be written as I s J da d dt v ( J ) dτ ( ) ρ J dτ ρdτ dτ v Charge is cnserved whatever flws ut must cme at the expense f that remaining inside - utward flw decreases the charge left in v J ρ This is called equatin f cntinuity Preparatry Schl t the Winter 0

22 quatin f Cntinuity Cnt d In Magnetstatic steady currents flw in the wire and its magnitude I must be the same alng the line- therwise charge wuld be pilling up sme where and current can nt be maintained indefinitely. ρ 0 In Magnetstatic and equatin f cntinuity J 0 Steady Currents: The flw f charges that has been ging n frever - never increasing - never decreasing. Preparatry Schl t the Winter 1

23 Magnetstatic and Current Bit and Savart Law μ I r dl 4π r ( ) 0 p B 3 Distributins dl is an element f length. r vectr frm surce t pint p. dl I r db p μ0 Permeability f free space. Unit f B N/Am Tesla (T) Preparatry Schl t the Winter

24 Bit and Savart Law fr Surface and Vlume Currents μ K r B 0 da 4π 3 r Fr Surface Currents μ J r B 0 dτ 4π 3 r Fr Vlume Currents Preparatry Schl t the Winter 3

25 Frce between tw parallel wires The magnetic field at () due t current I 1 is B μ π I 0 1 Pints inside 1 d Magnetic frce law df I ( ) dl B 1 μ I df I dl 0 1 kˆ π d Preparatry Schl t the Winter I1 I d df ( 1 ) ( ) dl B 1 4

26 Frce between tw parallel wires df μ 0 I 1 π I d dl The ttal frce is infinite but frce per unit length is df dl μ 0 I 1 π I d If currents are anti-parallel the frce is repulsive. Preparatry Schl t the Winter 5

27 Straight line currents The integral f B arund a circular path f radius s, centered at the wire is The current is ut f the page μ B dl 0 dl μ I 0 π s Fr bundle f straight wires. Wire that passes thrugh lp cntributes nly. B dl μ0 I I enc Applying Stkes therem B μ J 0 Preparatry Schl t the Winter 6

28 Divergence and Curl f B Bit-Savart law fr the general case f a vlume current reads μ ( ) 0 J r r dτ 4π r B 3 r r B 0 and B μ 0 J Preparatry Schl t the Winter 7

29 B Ampere s Law μ 0 J Ampere s law Integral frm f Ampere s law Using Stkes therem ( B) da B dl μ J da 0 B dl μ0 I enc Preparatry Schl t the Winter 8

30 Vectr Ptential The basic differential law f Magnetstatics B μ 0 J B 0 B curl f sme vectr field called vectr ptential A( P) B P A P μ ( ) ( ) ( ) J A 0 Culmb s gauge A 0 A μ J 0 Preparatry Schl t the Winter 9

31 Magnetstatic Field in Matter Magnetic fields- due t electrical charges in mtin. xamine a magnet n atmic scale we wuld find tinny currents. Tw reasns fr atmic currents. lectrns rbiting arund nuclei. lectrns spinning n their axes. Current lps frm magnetic diples - they cancel each ther due t randm rientatin f the atms. Under an applied Magnetic field- a net alignment f - magnetic diple ccurs- and medium becmes magnetically plarized r magnetized Preparatry Schl t the Winter 30

32 Magnetizatin If m is the average magnetic diple mment per unit atm and N is the number f atms per unit vlume, the magnetizatin is define as M Nm m Ia Am r m Mdτ Am A M m 3 m Preparatry Schl t the Winter 31

33 Magnetic Materials Paramagnetic Materials The materials having magnetizatin parallel t B are called paramagnets. Diamagnetic Materials The elementary mment are nt permanent but are induced accrding t Faraday s law f inductin. In these materials magnetizatin is ppsite t B. Ferrmagnetic Materials Have large magnetizatin due t electrn spin. lementary mments are aligned in frm f grups called dmain Preparatry Schl t the Winter 3

34 The Field f Magnetized Object Using the vectr ptential f current lp μ0 m rˆ A 4π r μ ˆ 0 M n μ0 M A da + d 4π r 4π r K ˆ b M n J M b τ Bund Surface Current Bund Vlume Current r Preparatry Schl t the Winter 33

35 Ampere s Law in Magnetized Material 1 μ 0 where Integral frm B μ 0 J J J b + J f Jb + J f J H J f B H M ( B) + ( M ) μ 0 H. dl I f fenc Preparatry Schl t the Winter 34

36 Faraday s Law f Inductin Faraday s Law - a changing -magnetic flux thrugh circuit induces an electrmtive frce arund the circuit. φ ε. d d dl B. da dt dt Є Induced emf Induced electric field intensity Differential frm f Faraday s law B t Preparatry Schl t the Winter 35

37 Faraday s Law f Inductin Induced lectric field intensity in terms f vectr ptential Fr steady currents A V V V Scalar ptential Induced emf in a system mving in a changing magnetic field B ε + v B ( ) Preparatry Schl t the Winter 36

38 Maxwell s quatins Preparatry Schl t the Winter 37

39 Intrductin t Maxwell s quatin In electrdynamics Maxwell s equatins are a set f fur equatins, that describes the behavir f bth the electric and magnetic fields as well as their interactin with matter Maxwell s fur equatins express Hw electric charges prduce electric field (Gauss s law) The absence f magnetic mnples Hw currents and changing electric fields prduces magnetic fields (Ampere s law) Hw changing magnetic fields prduces electric fields (Faraday s law f inductin) Preparatry Schl t the Winter 38

40 Histrical Backgrund 1864 Maxwell in his paper A Dynamical Thery f the lectrmagnetic Field cllected all fur equatins 1884 Oliver Heaviside and Willard Gibbs gave the mdern mathematical frmulatin using vectr calculus. The change t vectr ntatin prduced a symmetric mathematical representatin, that reinfrced the perceptin f physical symmetries between the varius fields. Preparatry Schl t the Winter 39

41 lectrdynamics Befre Maxwell Gauss s Law N name Faraday s Law Ampere s Law ( i) ρ ε ( ii) B 0 B ( iii) ( iv) B μ J A V B A Preparatry Schl t the Winter 40

42 lectrdynamics Befre Maxwell (Cnt d) Apply divergence t (iii) The left hand side is zer, because divergence ( ) B ( ) B f a curl is zer. The right hand side is zer because B 0. Apply divergence t (iv) ( ) ( ) B μ J Preparatry Schl t the Winter 41

43 lectrdynamics Befre Maxwell The left hand side is zer, because divergence f a curl is zer. The right hand side is zer fr steady currents i.e., (Cnt d) J In electrdynamics frm cnservatin f charge 0 J ρ t ρ t 0 ρ is cnstant at any pint in space which is wrng. Preparatry Schl t the Winter 4

44 Maxwell s Crrectin t Ampere s Law Cnsider Gauss s Law ε D t ( ) ε ρ ρ ε ε ρ t Displacement current This result alng with Ampere s law and the cnservatin f charge equatin suggest that there are actually tw surces f magnetic field. The current density and displacement current. Preparatry Schl t the Winter 43

45 Maxwell s Crrectin t Ampere s Law (Cnt d) Amperes law with Maxwell s crrectin B μ J + μ ε Preparatry Schl t the Winter 44

46 General Frm f Maxwell s quatins Differential Frm Integral Frm B B ρ ε 0 μ B J + μ ε S S C C d S B d S dl 1 ε 0 B d l μ I d dt V S enc ρdv B d S + μ ε d dt S d S Preparatry Schl t the Winter 45

47 Maxwell s quatins in vacuum The vacuum is a linear, hmgeneus, istrpic and dispersin less medium Since there is n current r electric charge is present in the vacuum, hence Maxwell s equatins reads as These equatins have a simple slutin interms f traveling sinusidal waves, with the electric and magnetic fields directin rthgnal t each ther and the directin f travel B B 0 0 B μ ε Preparatry Schl t the Winter 46

48 Maxwell s quatins Inside Matter Maxwell s equatins are mdified fr plarized and magnetized materials. Fr linear materials the plarizatin P and magnetizatin M is given by D B Where χ ε m μ is M ( 1 + χ ) ( ) H + M ( 1 + χ ) χ + e P the magnetic ε χ e χ m H the D and B fields are related PAnd t and H by e ε m ε μ H μh is the electric susceptibi lity f material, susceptibi lity f material and. Preparatry Schl t the Winter 47

49 Maxwell s quatins Inside Matter (Cnt d) Fr plarized materials we have bund charges in additin t free charges σ ρ b b P n Fr magnetized materials we have bund currents K J b b M n M P Preparatry Schl t the Winter 48

50 Maxwell s quatins Inside Matter (Cnt d) In electrdynamics any change in the electric plarizatin invlves a flw f bund charges resulting in plarizatin current J P J p P Plarizatin current density is due t linear mtin f charge when the lectric plarizatin changes Ttal ρ t Ttal ρ charge f + ρ current b density density J J + J + t f Preparatry Schl t the Winter b J p 49

51 Preparatry Schl t the Winter 50 Maxwell s quatins Inside Matter (Cnt d) Maxwell s equatins inside matter are written as t J J J B t B B b p f t ε μ μ μ μ ε ρ 0 ( ) D t J H P t J M B t M t P J B f f f ε μ ε μ

52 Maxwell s quatins Inside Matter (Cnt d) In nn-dispersive, istrpic media ε and µ are time-independent scalars, and Maxwell s equatins reduces t ε ρ μ H 0 μ H t H J + ε t Preparatry Schl t the Winter 51

53 Maxwell s quatins Inside Matter (Cnt d) In unifrm (hmgeneus) medium ε and µ are independent f psitin, hence Maxwell s equatins reads as D ρ H 0 H μ H J f f + ε S S D d S Q H d S 0 C C enc d dl μ dt H dl I f f enc S + H d S d dt S D d S Generally, ε and µ can be rank- tensr (3X3 matrices) describing birefringent anistrpic materials. Preparatry Schl t the Winter 5

54 Ptential Frmulatin f lectrdynamics 1 In electrstatic 0 V In electrdynamics But B B 0 A 0 Putting this in Faraday s Law V ( A) A xplain Maxwell s ii and iii equatins Preparatry Schl t the Winter 53

55 Ptential Frmulatin f lectrdynamics As V + V + A ρ ε A ρ ε ρ ε Pissn s quatin This replaces Pissn s quatin in electrdynamics Preparatry Schl t the Winter 54

56 Ptential Frmulatin f lectrdynamics 3 Preparatry Schl t the Winter 55 ( ) J t V A t A A t A t V J A μ ε μ ε μ ε μ ε μ μ + These equatin carry all infrmatin in Maxwell s equatins

57 The lectrmagnetic Waves Preparatry Schl t the Winter 56

58 lectrmagnetic Wave quatin The electrmagnetic wave equatin is a secnd-rder partial differential equatin that describes the prpagatin f electrmagnetic waves thrugh a medium r in a vacuum. T btain the electrmagnetic wave equatin in a vacuum we begin with the mdern 'Heaviside' frm f Maxwell's equatins. Preparatry Schl t the Winter 57

59 Frm Maxwell s quatins t the lectrmagnetic Waves 1 The Wave quatin 0 Maxwell s equatin in free space n charge r n current are given as B B 0 B μ ε Preparatry Schl t the Winter 58

60 Frm Maxwell s quatins t the lectrmagnetic Waves Take curl f B t Change the rder f differentiatin n the R.H.S [ B t ] [ B] Preparatry Schl t the Winter 59

61 Frm Maxwell s quatins t the lectrmagnetic Waves 3 As B Substituting fr B we have μ ε [ ] B [ ] [ μ ε ] [ ] μ ε As µ and ε are cnstant in time Preparatry Schl t the Winter 60

62 Frm Maxwell s quatins t the lectrmagnetic Waves 4 Using the vectr identity becmes, ( ) μ ε In free space 0 And we are left with the wave equatin μ ε 0 Preparatry Schl t the Winter 61

63 Frm Maxwell s quatins t the lectrmagnetic Waves 5 Similarly the wave equatin fr magnetic field B μ ε B 0 where, c 1 μ ε Preparatry Schl t the Winter 6

64 lectrmagnetic Wave quatin in Vacuum B μ ε 0 B μ ε 0 The slutins t the wave equatins, when there is n surce charge present can be plane waves - btained by methd f separatin f variables Preparatry Schl t the Winter 63

65 Slutin f lectrmagnetic Wave Plane electrmagnetic waves can be expressed as B 1 c e i ( k r ωt ) e i nˆ )( ( ) 1 ( ) k r ωt kˆ nˆ kˆ c nˆ Where is the plarizatin vectr and. kˆ is a prpagatin vectr. Preparatry Schl t the Winter 64

66 lectrmagnetic Plane waves Plane wave - a cnstant-frequency wave whse wave-frnts (surfaces f cnstant phase) are infinite parallel planes f cnstant amplitude nrmal t the phase velcity vectr. It is als used t describe waves that are apprximately plane waves in a lcalized regin f space. Fr example, a lcalized surce such as an antenna prduces a field that is apprximately a plane wave in its far-field regin. Preparatry Schl t the Winter 65

67 lectrmagnetic Plane waves The "rays" in the limit where ray ptics is valid (i.e. fr prpagatin in a hmgeneus medium ver length scales much lnger than the wavelength) crrespnd lcally t apprximate plane waves. Preparatry Schl t the Winter 66

68 Real lectrmagnetic Plane waves The real electric and magnetic fields in the frm f a mnchrmatic plane wave with prpagatin vectr kˆ and plarizatin nˆ ( ), t r cs( k r ω t) nˆ ( ) ( ) t k r t k n c B r, 1 cs( ω ) ˆ Preparatry Schl t the Winter 67

69 Hmgenus Wave quatins Inside Matter The hmgeneus frm f the equatin - written in terms f either the electric field r the magnetic field B - takes the frm: Vacuum Matter 1 μ ε 1 με μ 1 ε B B 1 με B B Preparatry Schl t the Winter 68

70 Hmgenus Wave quatins Inside Matter 1 Permittivity: εε r ε (ε r is dielectric cnstant) Permeability: µµ r µ (µ r is relative permeability v με μ μ r 0 ε ε r 0 με 0 0 με r r v c n c n nrefractive Index Preparatry Schl t the Winter 69

71 Plarizatin The plarizatin is specified by the rientatin f the electrmagnetic field. Histrically, the rientatin f a plarized electrmagnetic wave has been defined in the ptical regime by the rientatin f the electric vectr, and in the radi regime, by the rientatin f the magnetic vectr. Light is a transverse electrmagnetic wave. Natural light is generally un-plarized- all planes f prpagatin being equally prbable. Preparatry Schl t the Winter 70

72 Plarizatin Light in the frm f a plane wave in space is said t be linearly plarized. If light is cmpsed f tw plane waves f equal amplitude by differing in phase by 90, then the light is said t be circularly plarized. If tw plane waves f differing amplitude are related in phase by 90, r if the relative phase is ther than 90 then the light is said t be elliptically plarized. Preparatry Schl t the Winter 71

73 Linear Plarizatin In electrdynamics, linear plarizatin r plane plarizatin f electrmagnetic radiatin is a cnfinement f the electric field vectr r magnetic field vectr t a given plane alng the directin f prpagatin. The plane cntaining the electric field is called the plane f plarizatin. Preparatry Schl t the Winter 7

74 Linear Plarizatin Linear plarizatin can be hrizntal r vertical lectrmagnetic Wave lectric Field Magnetic Field Vertical Plarizatin y Hrizntal Plarizatin y x z x z Preparatry Schl t the Winter 73

75 Circular Plarizatin A plarizatin in which the tip f the electric field vectr, at a fixed pint in space, describes a circle as time prgresses. The electric vectr, at ne pint in time, describes a helix alng the directin f wave prpagatin. The magnitude f the electric field vectr is cnstant as it rtates. Circular plarizatin is a limiting case f the mre general cnditin f elliptical plarizatin. Preparatry Schl t the Winter 74

76 Circular Plarizatin Preparatry Schl t the Winter 75

77 lliptical Plarizatin In electrdynamics, elliptical plarizatin is the plarizatin f electrmagnetic radiatin such that the tip f the electric field vectr describes an ellipse in any fixed plane intersecting, and nrmal t, the directin f prpagatin. An elliptically plarized wave may be reslved int tw linearly plarized waves in phase quadrature- with their plarizatin planes at right angles t each ther. Preparatry Schl t the Winter 76

78 lliptical Plarizatin Preparatry Schl t the Winter 77

79 Preparatry Schl t the Winter 78 nergy and Mmentum f lectrmagnetic Waves The energy per unit vlume stred in electrmagnetic field is B U μ ε In the case f mnchrmatic plane wave ) ( cs 1 t kx U c B ω ε ε ε μ

80 nergy and Mmentum f lectrmagnetic Waves (Cnt d) As the wave prpagates, it carries this energy alng with it. The energy flux density (energy per unit area per unit time) transprted by the field is given by the pynting vectr S μ 1 ( ) B Fr mnchrmatic plane waves S cε cs ω ( kx t) iˆ cui ˆ Preparatry Schl t the Winter 79

81 References 1. CLASSICAL LCTRODYNAMICS By J. D. Jacksn (WILY). INTRODUCTION TO LCTRODYNAMICS By David. J. Griffiths ( PRNTIC HALL) Preparatry Schl t the Winter 80

82 THANK YOU Preparatry Schl t the Winter 81