Lecture #5.1 Electromagnetic Waves During our study of classical mechanics we discussed mechanical waves. The most important everyday phenomenon to

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1 Leture #5. Eletromageti Waves Durig our study of lassial mehais we disussed mehaial waves. The most importat everyday pheomeo to whih you a apply your kowledge of mehaial waves is distributio of soud. We leared that harmoi waves give a example of periodi proess i both time ad i spae. Mehaial wave is ot the oly example of the wave motio i ature. Aother pheomeo whih has remarkably similar behavior is light. Light, of ourse, is ot a mehaial wave. I order for mehaial wave to propagate, it is eessary to have some sort of medium. We kow, however, that we a see light omig from distat stars. They are separated from us by a essetially empty spae, whih a ot trasmit ay mehaial waves. So, the ature of light must be differet. The first aswer whih omes i mid is that maybe light is ot a wave at all. To see if this is true or ot, we have to study properties of light i details. First of all let us try to uderstad what eletromageti wave is. We already kow the basi properties of eletri ad mageti fields. The first oditio for eletri field to exist is the presee of eletri harges. We kow that a statioary eletri harge produes eletrostati field. At the same time movig eletri harge produes mageti field. If this harge is ot just movig with ostat speed but is aeleratig the we have a example of alteratig eletri urret. This urret will be a soure of hagig mageti field. So, if oe osiders mageti flux of this hagig mageti field through the surfae of some losed loop the, aordig to Faraday s law, the eletromotive fore or idued eletri field will appear i this loop. So we see a patter: hagig eletri field auses appearae of the hagig mageti field whih i its tur auses the appearae of the hagig eletri field ad so o. This is how eletromageti waves are formed. This proess a be desribed mathematially with the help of Maxwell s equatios. Theoretial preditios made by Maxwell o the basis of these equatios were prove i real experimets. Solvig the equatios i geeral, Maxwell obtaied the solutio i the form of the eletromageti wave whih, aordig to his theory, should propagate i vauum with the speed of light. Sie eletromageti wave, as ay other wave, is the proess of propagatio of eergy i spae the this eletromageti wave a be deteted by plaig the soure of the wave (trasmittig atea) i oe poit ad detetig the wave by

2 meas of the reeivig atea i the other poit i spae. Those experimets were first performed by Hertz. Eletromageti (EM) wave is a trasverse wave, where both eletri ad mageti fields are i the diretio perpediular to the diretio of the wave s trasmissio. At the same time eletri field is perpediular to mageti field. If we have harmoi si-like eletromageti wave, whih is the simplest example of suh a wave, both eletri ad mageti fields are hagig with time aordig to the si-like law. It turs out that those hages for both fields are i phase, so both eletri ad mageti fields are reahig their maximum ad miimum values simultaeously. The maximum values of the fields are related as E B (5.5.) Diretio of propagatio of EM wave is related to diretios of eletri ad mageti fields by meas of the same right-had rule: Poit the figers of your right had i the diretio of eletri field, url your figers towards diretio of mageti field ad your thumb will poit i the diretio of propagatio of the wave. Sie eletromageti waves are related to behavior of both types of fields, Maxwell was able to alulate the speed of those waves from his equatios i terms of 8 both eletri ad mageti ostats, whih is 3*0 m s. O the other had oe a measure the speed of light usig Fizeau s experimet with rotatig wheel. This experimet predits the same value as the speed of eletromageti wave i vauum from Maxwell s equatios. The exat value of the speed of light, based o the moder 8 measuremets is m s. Eve though all eletromageti waves have the same speed i vauum (about the same as i air) but they a have differet wavelegths ad frequeies. We kow how wavelegth, frequey ad speed are related, whih is f. (5..) So, the higher the frequey is the lower wavelegth will be. You a estimate wavelegths of typial radio-waves by kowig frequey of radio statio. Visible light has frequeies muh higher ad wavelegths muh shorter tha radio waves. Figure 33. i the book shows values of wavelegths ad frequeies for differet parts of eletromageti spetrum. You a see that visible light of differet olors has differet wavelegths. The etire visible spetrum is loated i the rage of wavelegths from 0 0

3 m for violet olor to m for red olor. The waves with lower wavelegths are alled ultraviolet light; the waves with higher wave legths are alled ifrared light. Eve though all eletromageti waves have the same speed i vauum, but they may have very differet speeds i differet media. For istae, we kow that walls of the buildig do ot allow passig of visible light but radio waves a peetrate ito the buildig. I ay material the speed v of eletromageti wave is goig to be less ompared to its value i vauum, whih is (5..3) v idex of refratio for a give substae is always bigger tha. Not oly this idex of refratio depeds o the type of material, but it also depeds o frequey of EM wave. I the ase of visible light, the light of differet olors will have differet speeds i the same medium. This pheomeo is kow as dispersio. As light passes from oe substae to aother substae it should have the same frequey, beause atoms of the ew substae are goig to osillate at the same frequey as the frequey of EM wave, whih auses these osillatios. This meas that havig the same frequey but differet speed, light should have differet values of wavelegths i differet substaes. As light travels from vauum ito some other medium, its wavelegth dereases by the value of the idex of refratio for this medium. If oe direts a beam of white light to the glass prism, it will be beded by this prism. Beause wavelegths of differet olors have differet speeds i the medium of the prism, they will be baded by differet agles. As a result the white light will be split ito a spetrum of differet olors. This shows that white light is ot a pure light but rather a mixture of differet olors. This experimet was first performed by Newto. The huma eye is the most sesitive for wavelegths of three primary olors red, blue ad gree. All other olors whih we a see are the results of ombiatio of these 3 primary olors. However, the thigs we see i most part of ases do ot produe their ow light waves. Istead they reflet light omig from other soures of light. The fat that we see most of the thigs i differet olors meas that they reflet differet wavelegths differetly. The light of some olor (or olors) may be refleted by the objet ad so we see it appearig i this olor, while other olors are absorbed by this objet.

4 Sie eletromageti wave as ay other wave is a proess of propagatio of eergy i spae, we eed to figure out what is the eergy trasmitted by EM waves. This eergy is stored i the form of both eletri ad mageti eergy. At every poit the desity of eletri eergy is exatly the same as the desity of mageti eergy ad eah of them ompose the half of the wave s eergy. Durig our study of apaitors ad idutors, we have disussed both these types of eergy. The volume desity of eletri eergy is ue 0E (5..4) ad the volume desity of mageti eergy is ub B, (5..5) 0 so the total eergy desity of EM wave is u E B E B. (5..6) Sie both fields are osillatig with time, we a also itrodue rms ad average values of those quatities, whih is E max Erms, B max Brms, u E B avg 0 rms rms 0 (5..7) I a same way as for ay other type of waves, we a talk about wave s itesity, whih is the average eergy trasmitted by wave per uit of time through the uit of area perpediular to the diretio of the wave s propagatio. P u A t avg I uavg 0Erms Brms A A t 0 (5..8) Aother property of light whih we shall disuss is alled polarizatio. We have already metioed that light is a trasverse wave, where diretios of both eletri ad mageti fields are perpediular to the diretio of the wave s propagatio. To desribe this wave, we a speify the diretio i whih eletri field osillates. If eletri field osillates i oe plae oly the this will be a example of ompletely polarized light. I reality, however, regular light is a mixture of waves with differet diretios of

5 polarizatio. This is alled upolarized light. The most ommo way to produe polarized light from upolarized light is by usig polarizatio filter whih oly allows waves with ertai polarizatio to pass. Suh filters a be made from diahroi rystals or from polymer films (for istae polymer films are plaed o glasses ad photo objetives). I order to see if the light is polarized or ot oe a put two filters with differet orietatios of the polar axes. The first filter will produe the light polarized i ertai diretio. If the filter s trasmissio axis makes agle with the diretio of eletri field i EM wave, the the oly ompoet of eletri field, whih a pass through the filter is the ompoet of the field parallel to its axis, whih is E Eos. Rememberig that itesity of light is proportioal to the square of eletri field, we a olude that the itesity of light passig through this filter is goig to be I I os, (5..9) 0 where I 0 is the origial itesity of upolarized light. This equatio is kow as Malus law. If origial light was ompletely upolarized (with equal probability of havig ay orietatio of the eletri field) the it will loose half of its itesity passig through a polarizer. I I0 (5..0) If oe uses filters the rotatig the seod filter relative to the first filter we a ahieve the situatio whe o light is passig through this ombiatio at all. This happes if the axis of the seod filter is perpediular to the axis of the first filter. The fat that light is a wave was prove by studyig differet pheomea suh as iterferee, diffratio ad polarizatio. We will talk about iterferee ad diffratio later but we have just see that polarizatio does take plae for light as well as for ay other EM wave. This pheomeo a oly be explaied ad uderstood if we admit wave ature of light. However, there are may other thigs we kow about light ad use i our everyday life, whih a be explaied without lookig ito its wave ature. I fat, most of these pheomea were desribed before people eve realized that light is a wave. This refers to thigs suh as formatio of images i mirrors ad simple leses. As log as the size of the objet is large eough ompared to the wavelegth of light, it is ot eessary to aout for its wave properties. The part of physis whih studies light from this stadpoit is kow as Geometrial Optis. The oly assumptio about light that we will

6 have to make i order to desribe these pheomea is that light propagates alog the straight lies. If we wat we a look at this from the wave-theory stadpoit. We kow that light waves have wave frots. The wave frots are the surfaes where eletromageti wave has a ostat phase. I the ase of the poit-like soure of light, we have a spherial wave propagatig i all diretios from the soure. So, the wave frots are spheres ad light s rays are the straight lies i the radial diretios from the soure. If the distae from the soure is relatively large, we a eglet urvature of the wave frot ad osider light as a plae wave or as a straight ray goig i horizotal diretio. This is the oly assumptio we eed to study the proesses related to refletio ad refratio of light rays. Talkig about image formatio i geeral, we kow, of ourse, that ot all the objets are soures of light. For istae, if you tur off the lights i the room without widows, you will hardly be able to see aythig. So our visio beomes possible oly beause we reeive seodary light rays. These rays are first emitted by a soure of light ad the they are refleted from the objets we see. This meas that i order to uderstad image formatio, we first have to uderstad how refletio takes plae. The reaso for this refletio is the fat that every poit o the surfae of the objet, whih reflets light rays, beomes a soure of the seodary light rays. Those rays are also emitted i all radial diretios as it was i the ase of primary light s soure. The simplest example is the refletio from the perfet plae mirror. Suppose the ray of light hits the mirror at some agle relative to the ormal. This agle betwee the diretio of the light ray ad the ormal lie to the mirror is alled the agle of iidee. Sie, i geeral, this agle is ot zero the differet parts of the wave frot will hit the mirror at differet momets. By the time whe the last part of the wave frot reahes the mirror, the first part of the wave frot is already refleted. It oly beomes possible if the agle whih the refleted ray makes with the ormal to the mirror (alled the agle of refletio) is the same as agle of iidee. This is kow as law of refletio, whih is that for a smooth refletig surfae, the agle of iidee is equal to the agle of refletio (5..) i r Also the refleted ray is loated i the same plae as the iidet ray. This law allows us to explai how the images are formed by a plae mirror.

7 Refletio is ot the oly physial pheomeo, whih may our at the boudary betwee the two differet substaes. If the substae is a odutor (some metalli substae) the eletri field aot exist iside of this odutor ad refletio ours. However, if the substae is ot a odutor but some other material, whih allows trasmittig of light (for istae glass), part of light will be refleted but aother part will peetrate ito the ew medium. Let us osider a plaar border betwee the air ad a glass. The light ray travels i the air ad hits the glass surfae at some o zero agle of iidee, the it will otiue to propagate i glass. However, the speed of light i glass is less tha the speed of light i the air. Sie the ray omes to the glass surfae at some agle, differet parts of the wave frot will hit the surfae at differet momets, so some part of the wave frot will be already iside of glass ad travel slowly, while the other part is still i the air ad travels relatively fast. As a result, the light ray will tur to the ertai agle. This pheomeo is kow as refratio. The magitude of refratio agle depeds o how may times the speed of light i the medium is differet ompared to the speed of light i the air, whih is (aordig to equatio 5..3) the idex of refratio. Whe the ray passes ito glass, it makes agle (alled the agle of refratio) relative to the ormal to the boudary betwee the two media. This agle is ow differet ompared to the agle of iidee. Those two agles are related aordig to si v f v, (5..) si v f v whih is also kow as the Shell s law si si, (5..3) where, are stadig for idies of refratio of eah of the two materials (air ad glass i my example). More dese substaes usually have larger idies of refratio, so the light passes from the less dese substae ito more dese substae ad if light passes from more dese substae ito less dese substae. Oe agai the iidet ray, the ormal to the boudary surfae ad the refleted ray are i the same plae. You are to verify this law i the lab tomorrow. Usig this law oe a explai why the objets appear loser to the surfae of water, whe see from above this surfae, tha they atually are.

8 The iterestig pheomeo ours tha the light travels from the deser medium with higher idex of refratio ito less dese medium with lower idex of refratio. I this situatio ad at some poit agle reahes its ritial value si si90 o, (5..4) suh that refrated ray just skims the surfae of the seod medium. If the agle of iidee is larger tha this ritial value, the total iteral refletio ours ad the ray omes bak to the origial deser medium. I the ase of the glass with idex of refratio.5, this ritial agle of iidee is 4 degrees, so the glass-air boudary will be a perfet mirror for rays omig from glass at agles of iidee higher tha 4 degrees. The pheomeo of total iteral refletio is used i differet optial devies, suh as bioulars, where prisms are used to fold light by meas of total iteral refratios. Aother appliatio is the optial fiber used for trasmissio of light sigals. Aother iterestig fat is that polarizatio of light ours durig total or partial refletio from the dese ometalli substae. There is oe ase, whe the refleted light is ompletely polarized. That takes plae if the agle betwee refrated ad refleted rays is 90 degrees the the diretio of total polarizatio is parallel to the refletig surfae. The iidee agle, whe this pheomeo ours is alled Brewster s agle. For this agle we have si si, B o 90, B si si 90 os, o B B B ta B (5..5) For the typial situatio, whe the first medium is air ad the seod medium is glass with idex of refratio.5, the Brewster s agle of iidee is about 56 degrees.