Lecture 17. Phys. 207: Waves and Light Physics Department Yarmouk University Irbid Jordan

Transcription

1 Leture 17 Phys. 7: Waves and Light Physis Departent Yarouk University 1163 Irbid Jordan Dr. Nidal Ershaidat Maxwell s Equations In 187, Jaes Clerk Maxwell's established a atheatial frae based on four equations in eletriity and agnetis, in whih all phenoena eletriity and agnetis an be explained. Eletriity and agnetis were thus unified in what we all now Eletroagnetis. 1

2 Maxwell s Equations E. d A B. d A q ε Gauss Law for eletriity Gauss Law for agnetis 3 E. d B.d l l dφ dt B µ i + ε µ dφ dt Faraday s Law E Apere-Maxwell Law Light is e Waves Jaes Clerk Maxwell's showed that eletroagneti energy propagates in vauu with the speed of light (). His ajor onlusion was that light is nothing else but e waves, i.e. a bea of light is a traveling wave of eletri and agneti fields an eletroagneti wave Optis, the study of visible light, beae a branh of eletroagnetis. We shall onentrate on stritly eletri and agneti phenoena, and we build a foundation for optis. 4

3 and MKS 5 ε 1 µ The MKS syste (SI) has for base this equation. All other units are defined starting fro the definition of. H. Hertz 6 In Maxwell's tie, the visible, infrared, and ultraviolet fors of light were the only eletroagneti waves known. Heinrih Hertz disovered what we now all radio waves and verified that they ove through the laboratory at the sae speed as visible light. 3

4 The Eletroagneti Spetru 1 n Å Note the overlap between types of waves Visible light is a sall portion of the spetru Types are distinguished by frequeny or wavelength Fig. 1 Visible Light The relative sensitivity of the average huan eye to eletroagneti waves at different wavelengths. This portion of the eletroagneti spetru to whih the eye is sensitive is alled visible light. 8 Fig. 3 4

5 5- The Traveling Eletroagneti Wave, Qualitatively The Eletroagneti Spetru 1! Any aelerating harge will eit eletroagneti waves.!an eletroagneti wave is a wave that obines the eletri wave eletri field E and agneti wave agneti field B.! Osillating harges is an exaple aelerating harges. 5

6 Generation of eletroagneti waves An LC iruit is used to produe e osillations. An energy soure is used to opensate for the energy loss in the wires (represented by the resistane R) Osillations are transitted to an antenna whih eits the e waves in spae. 11 Fig. 4 Propagation of an e wave Desription of the eletri and agneti fields as they pass point P on the Fig.4 1 Propagation diretion of the wave Fig. 5 6

7 Properties of the eletri and agneti field in eletroagneti waves. EM Waves are transverse waves 1. EM waves are transverse waves. The eletri and agneti fields E r and B r are always perpendiular to the diretion of travel of the wave., as disussed in Chapter 17..The eletri field is always perpendiular to the agneti field. 3.The ross produt E r B r always gives the diretion of travel of the wave. 14 7

8 15 Properties of the eletri and agneti field in eletroagneti waves. 4.The fields always vary sinusoidally, just like the transverse waves disussed in Chapter 17. Moreover, the fields vary with the sae frequeny and in phase (in step) with eah other. 5.The E and B field an be desribed by: E ( x, t ) B( x, t ) E B sin( k x ω t ) sin( k x ω t ) Speed of propagation 6. All eletroagneti waves, inluding visible light, have the sae speed in vauu Where is given by: E E 1 B B µ ε 16 8

9 Speed in Material Media 7. Speed of light in dense ediu is less than : v n 17 Where n is a paraeter whih haraterizes the ediu and alled the index of refration of the ediu. Leture 18 Phys. 7: Waves and Light Physis Departent Yarouk University 1163 Irbid Jordan Dr. Nidal Ershaidat

10 5- The Traveling Eletroagneti Wave, Qualitatively The Eletroagneti Spetru! Any aelerating harge will eit eletroagneti waves.!an eletroagneti wave is a wave that obines the eletri wave eletri field E and agneti wave agneti field B.! Osillating harges is an exaple aelerating harges. 1

11 Generation of eletroagneti waves An LC iruit is used to produe e osillations. An energy soure is used to opensate for the energy loss in the wires (represented by the resistane R) Osillations are transitted to an antenna whih eits the e waves in spae. 1 Fig. 4 Propagation of an e wave Desription of the eletri and agneti fields as they pass point P on the Fig.4 Propagation diretion of the wave Fig. 5 11

12 EM Waves fro an Antenna Two rods are onneted to an a soure, harges osillate between the rods (a) As osillations ontinue, the rods beoe less harged, the field near the harges dereases and the field produed at t oves away fro the rod (b) The harges and field reverse () The osillations ontinue (d) 3 Fig. 6 Propagating Osillations 4 Current (up and down) reates B field into and out of the page! + y - z x Fig. 7 1

13 Properties of eletroagneti waves. EM Waves are transverse waves 1. EM waves are transverse waves. The eletri and agneti fields E r and B r are always perpendiular to the diretion of travel of the wave, as disussed in Chapter

14 Speed of propagation. All eletroagneti waves, inluding visible light, have the sae speed in vauu Where is given by: E E 1 B B µ ε 7 v λ f ω k Fig. 8 Speed in Material Media 3. Speed of light in dense ediu is less than : v n 8 Where n is a paraeter whih haraterizes the ediu (the index of refration of the ediu). 14

15 Properties of the eletri and agneti field in eletroagneti waves. 1.The eletri field is always perpendiular to the agneti field..the ross produt E r B r always gives the diretion of travel of the wave. 3.The fields always vary sinusoidally, just like the transverse waves disussed in Chapter 17. Moreover, the fields vary with the sae frequeny and in phase (in step) with eah other. 4.The E and B field an be desribed by: E ( x, t ) E sin( k x ω t ), B( x, t ) B sin( k x ω t ) 9 Plane Eletroagneti waves. 3 The previous equations define what we all a plane wave E ( x, t ) E sin( k x ω t ), B( x, t ) B sin( k x ω t ) i ( k x ωt ) e Whih we an write in a oplex for as: E i ( ) ( ) k x ω t i ( k x ω t ) x, t E e, B( x, t ) B e 15

16 5-3 The Traveling Eletroagneti Wave, Quantitatively E /B - Indued Eletri Field Fig. 9 shows a wavefront at soe referene instant t. After tie dt, the wavefront has traveled dt. Consider a representative plane of area A. E r propagates in the y + diretion and B r in the z + diretion 3 Fig. 9 16

17 E /B, Indued Eletri Field Let us apply Faraday s law of indution along the dashed path: dφ E + de B E. d l dt ( E + de) h E h hde E. d l d Φ B d( B A ) db A dt dt dt db hde hdx dt de db dx dt db hdx dt E dx h 33 E /B Osillations are desribed by: E ω de dx E db dt k os E B ω k ( x, t ) E sin( k x t ) ( x, t ) B sin( k x t ) B ω ( x, t ) E x B x, t t ( ) k E ( k x ω t ) + B ω os( k x ω t ) os ( k x ω t ) ( k x ω t) ω B os 34 17

18 E /B, Indued Magneti Field Let us apply Apere-Maxwell law of indution along the dashed path: B. d l ε µ B. d l d Φ E d( E A ) dt dt hdb ε µ db dx ε µ dφ dt E ( B + db) h B h hdb de hdx dt de dt de A dt de hdx dt B dx h E B + d B 35 E /B Osillations are desribed by: E ω db dx E x, t ω E os k x ω t t B( x, t ) k B sin( k x ω t ) x de ε µ dt B k os E 1 1 B ε µ ω k ε µ ( x, t ) E sin( k x t ) ( x, t ) B sin( k x t ) ( ) ( ) B ω ( k x ω t ) + ε µ E ω os( k x ω t ) ( ) 36 18

19 ε µ 1 We have: E B E B ε 1 ε µ µ Whih gives: 1 The speed of e waves is: 1 ε µ Energy Transport and the Poynting Vetor 19

20 Poynting Vetor Fro sun light, We all know that an eletroagneti wave an transport energy and deliver it to a body on whih it falls. The rate of energy transport per unit area in suh a wave is desribed by a vetor S r, alled the Poynting vetor. It is alled after physiist John Henry Poynting ( ), who first disussed its properties. 39 Poynting Vetor The diretion of the Poynting vetor S r of an eletroagneti wave at any point gives the wave's diretion of travel and the diretion of energy transport at that point. The Poynting vetor is defined by: 1 S E B µ S E 1 µ E, B and S B r. h. s The agnitude of the Poynting vetor is given by: 1 S E B µ 4

21 Power eitted and Poynting The units of S is power per unit area or W/ Its agnitude S is related to the rate at whih energy is transported by a wave aross a unit area at any instant. [ ] [ Energy ][ Tie ] [ Power ] S [ Area ] [ Area ] 41 Reeber that for a plane wave: E E B B E S ε E or S µ µ B Intensity Eitted 4 The intensity is defined as: I S avg 1 µ E avg E µ E rs µ or: I B µ 1 ε E E rs µ 1

22 Energy Moentu of an e Wave 43 The energy of eletroagneti waves is related to its oentu p by the following relation: U p If an eletroagneti wave is inident norally on an area A as shown in fig. 1 in a tie t and it is totally absorbed then there is oentu hange given by: U p A Where this oentu hange is totally transferred to the area A Fig. 1 Radiation Pressure If the eletroagneti wave is totally refleted then the hange in oentu is: U p The net fore on the wall is: 44 F p t ( 1or ) ( 1or ) I A t ( 1or ) U t The radiation pressure P r is: F P r A I ( 1or ) t I A

23 Leture 19 Phys. 7: Waves and Light Physis Departent Yarouk University 1163 Irbid Jordan Dr. Nidal Ershaidat Light Inident on an Objet 3

24 Light inident on an objet 47 Absorbed Reflets (bounes) Refration (bends) Law of Refletion 48 Angle between light bea and noral Angle of inidene Angle of refletion θ i θ f θ i θ f 4

25 Refrative index n and wavelength 49 Snell s Law When light travels fro one ediu to another the speed hanges v/n, but the frequeny is onstant. So the light bends: n 1 sin(θ 1 ) n sin(θ ) 5 θ 1 n 1 n θ 5

26 Snell s Law Exaple 1 51 Whih is true? 1) n 1 > n ) n 1 n 3) n 1 < n θ 1 < θ n 1 sin θ 1 < sin θ θ 1 n n 1 > n θ Snell s Law Exaple A ray of light traveling through the air (n1) is inident on water (n1.33). Part of the bea is refleted at an angle θ r 6. What is θ? θ 1 θ r θ 1 θ r 6 sin(6 ) 1.33 sin(θ ) 5 noral n 1 1. n 1.33 θ 4.6 degrees n sin(θ ) n sin(θ ) θ 6

27 Total Internal Refletion Snell s Law: n 1 sin(θ 1 ) n sin(θ ) when n 1 > n, θ > θ 1 When θ 1 sin -1 (n /n 1 ) θ 9 This is a ritial angle! 53 θ n Light inident at a larger angle will be opletely refleted θ i θ r θ i θ θ 1 θ r n 1 noral Total Internal Refletion 54 Total internal refletion an our when light attepts to ove fro a ediu with a high index of refration to one with a lower index of refration. Ray 5 shows internal refletion 7

28 Critial Angle A partiular angle of inidene will result in an angle of refration of 9 This angle of inidene is alled the ritial angle 55 sin θ n n 1 for n 1 > n Chroati Dispersion The index of refration n enountered by light in any ediu exept vauu depends on the wavelength of the light. The dependene of n on wavelength iplies that when a light bea onsists of rays of different wavelengths, the rays will be refrated at different angles by a surfae; that is, the light will be spread out by the refration. This spreading of light is alled hroati dispersion, in whih hroati refers to the olors assoiated with the individual wavelengths and dispersion refers to the spreading of the light aording to its wavelengths or olors 56 8

29 Chroati Dispersion 57 n1 sin θ sin θ n 1 Chroati Dispersion 58 red blue 9

30 Chroati Dispersion 59 Leture Phys. 7: Waves and Light Physis Departent Yarouk University 1163 Irbid Jordan Dr. Nidal Ershaidat 3

31 Proble 17 The axiu eletri field at a distane of 1 fro an isotropi point light soure is. V/. What are: a) the axiu value of the agneti field and b) the average intensity of the light there? ) What is the power of the soure? Solution: E 8 a) B Tesla b) I E µ π W / ) P 4π r I 4 π W Proble 5 A plane eletroagneti wave, with wavelength 3., travels in vauu in the positive x diretion with its eletri field E, of aplitude 3 V/, direted along the y axis (a) What is the frequeny f of the wave? f λ (b)what are the diretion and aplitude of the agneti field assoiated with the wave? B E Hz along the z axiz 6 31

32 Proble 5 ()What are the values of k and ω if E E sin(k x ω t)? k π λ π 3. 1 ω π f 8 π 1, rad s (d)what is the tie-averaged rate of energy flow in watts per square eter assoiated with this wave? I S E ( µ ) avg ( 3 ) ( π 1 ) W 63 Proble 5 (e)if the wave falls on a perfetly absorbing sheet of area., at what rate is oentu delivered to the sheet and what is the radiation pressure exerted on the sheet? dp I A N dt p r I N 64 3

33 Proble 39 A bea of partially polarized light an be onsidered to be a ixture of polarized and unpolarized light. Suppose we send suh a bea through a polarizing filter and then rotate the filter through 36 while keeping it perpendiular to the bea. If the transitted intensity varies by a fator of 5. during the rotation, what fration of the intensity of the original bea is assoiated with the bea's polarized light? 65 Proble 39 Solution Let I be the intensity of the inident bea and f the fration of it whih is polarized. Then the intensity of the polarized portion is fi : and the intensity of the unpolarized portion is (1-f) I : Initially the intensity of transitted portion before rotating the polariod is: I ( 1 f ) I + f I os θ The iniu intensity is : Iin I ( os θ ) ( 1 f ) I + { } ( 1 f ) I The axiu intensity is: ( θ 1) ( 1 f ) I + { f I } ( 1 f ) I I ax I os + Iax 1 + f 5 f I 1 f 3 in 66 33

34 Proble 49 Prove that a ray of light inident on the surfae of a sheet of plate glass of thikness t eerges fro the opposite fae parallel to its initial diretion but displaed sideways, as in Fig Show that, for sall angles of inidene θ, this displaeent is given by n 1 x t θ n where n is the index of refration of the glass and θ is easured in radians. Proble 59 In Fig. 34-5, light enters a 9 triangular pris at point P with inident angle θ and then soe of it refrats at point Q with an angle of refration of 9 (a) What is the index of refration of the pris in ters of θ? (b) What, nuerially, is the axiu value that the index of refration an have? Explain what happens to the light at Q if the inident angle at Q is () inreased slightly and (d) dereased slightly

35 Next Leture Chapter 6: Interferene 35