PLK VICWOOD K.T. CHONG SIXTH FORM COLLEGE Form Seven AL Physics: Wave Phenomena

Transcription

1 L Physics/Wave Phenomena/P. PLK VICWOOD K.T. CHONG SIXTH FORM COLLEGE Form Seven L Physics: Wave Phenomena Wave phenomena Huygens principle Reflecion Refracion Polarizaion Familiariy wih ripple ank experimens is assumed from lower form work. Explanaion of laws of reflecion and refracion. Examples o include brief discussion of radar, sonar and long disance propagaion of radio waves by reflecion from he ionosphere. Phase change on reflecion, illusraed for example, using a slinky spring. Refracion as a resul of change in wave speeds. Refracive index in erms of speeds. Polarizaion by selecive absorpion, reflecion and scaering. Pracical applicaions o include polaroid specacles, VHF and UHF anennas (briefly). E6. Polarising ligh by (a) reflecion from a shiny surface; (b) absorpion using a shee of polaroid; and (c) scaering using cloudy waer (NP 8.5/J 2d). Superposiion Mahemaical reamen no required. E7. Superposiion of ransverse waves on a slinky spring. Beas Qualiaive reamen. Use in uning. E8. Observaion of beas on a CRO. Diffracion Inerference Diffracion of ligh a aperures (simple qualiaive reamen only). Two-source inerference wih quaniaive reamen for maxima and minima. Condiions for observable inerference. Pracical applicaions of inerference o include he blooming of lenses and he esing of he flaness of a surface (very briefly). Quaniaive reamen of inerference effecs a normal incidence in parallel-sided and wedge-shaped hin films. Every day examples o include he colours of oil films and soap bubbles. Newon s rings (qualiaively). Plane ransmission graing as an inerference sysem. Use of he formula d sin θ = nλ. Proporionaliy beween inensiy and square of he ampliude (by analogy wih harmonic oscillaor and energy delivered by an alernaing curren). Energy disribuion in inerference paerns. E9. Looking a a lamp hrough a sli or a pin-hole o sudy how he diffracion paerns depend on (a) he shape of he aperure; (b) he size of he aperure; and (c) he wavelengh of ligh. (NP 8.a/J ) E0. Esimaion of he wavelengh of ligh using (a) double sli, and (b) plane diffracion graing. (Na Phil Workbook 5) E. Observaion of Newon s rings and inerference fringes in soap film. E2. Invesigaion of he ampliude and energy disribuion in an inerference paern of sound waves (NP 8.7/J 4).. Huygen s Principle - enables he new posiion of a wavefron o be found, knowing is posiion a some previous insan. - can be used o explain he reflecion, refracion, ec. - Huygen s Principle: every poin on a wavefron may be regarded as a source of secondary spherical waveles which spread ou wih he wave velociy. The new wavefron is he envelope of hese secondary waveles, i.e. he surface which ouches all he waveles.

2 L Physics/Wave Phenomena/P.2. Propagaion of sraigh/plane waves - Given he wavefron of a plane wave in free space, where will he wavefron be a ime laer? Secondary source c c c C Consruced wavefron rays showing he direcion of propagaion Firs posiion of wavefron c B D Secondary wavefron 2. Propagaion of circular/spherical waves C B D 3. Law of Reflecion - he wavefron B is inciden obliquely on he reflecing surface and has jus reached i. - o find he new posiion of he wavefron when B is abou o be refleced a B, - a secondary wavele wih cenre and radius BB is drawn. - The angen B from B o his wavele is he required refleced wavefron. In B and BB B = BB = 90º B is common = BB (by consrucion) herefore, B BB (RHS) BB = B i.e. i = r (Law of reflecion) inciden wavefron C i i B Secondary wavele from ' Refleced wavefron D r r B' 4. Law of Refracion - he end of a plane wavefron B abou o cross he boundary beween media! and " in which is speeds are v and v 2 respecively.

3 L Physics/Wave Phenomena/P.3 - The new posiion of he wavefron a ime laer when B has ravelled a disance BB = v and reached B is found by drawing a secondary wavele wih cenre and radius = v 2. - The angen B from B o his wavele is he required wavefron and he ray, which is normal o he wavefron, is a refraced ray. v BB sin i = B' Inciden wavefron B v 2 B sin i2 = C N B' i Hence, sin i v c/ v n v 2 2 i = = = (Snell s Law) v sin i2 v 2 c/ v n > v 2 i v 2 2 i where n and n 2 are absolue refracive index of he 2 media. N' ' ** Refracion occurs when a wave passes from one Secodnary wavele from medium ino anoher in which i has a differen speed. B'! " Refraced wavefron B. Reflecion. pplicaion of Reflecion a. Radar (Radio deecion and ranging) - radio waves (EM wave, λ: 30 km - 3 cm) are emied in shor pulses by a ransmier and picked up afer reflecion from he objec. - By measuring he ime beween he ransmied pulse and he refleced pulse, disance from he ransmiing saion o he objec is hen found. b. Sonar (Sound navigaion and ranging) - similar o radar bu employs ulrasonic waves (Sound waves wih frequency above 20 khz). - measure he deph of he sea, deecing shoals of fish, ec. c. Reflecion from he ionosphere - sreching from abou 80 km above he earh o 500 km - ionosphere, posiively charged hrough he removal of elecrons by he sun s ulraviole radiaion. - E.M. wave wih frequency below 30 MHz canno penerae he ionosphere bu refleced. - Shor waves (SW): 6 o 20 MHz bounced from he ionosphere and are widely used for long disance broadcass.

4 L Physics/Wave Phenomena/P.4 2. Phase Inversion a. - Consider a long narrow spring wih he righ-hand end of he spring being fixed. ransverse upward pulse ravels owards he fixed end and is refleced. Observaion: The refleced pulse is invered relaive o he inciden pulse!! i.e. a phase change of 80º or π rad has occurred. fixed end - now he righ-hand end of he spring becomes free, he ransverse upward pulse ravelling owards i is refleced. Observaion: The whole of he inciden pulse is refleced he righ way up. i.e. no phase change occurs!! -Why?? free end b. Principle of Superposiion When wo waves disurbances mee, he resulan displacemen a any poin is he vecor sum of he separae displacemens due o he wo waves. (i) (ii) (iii) (iv) (v) (a) Cres and cres (b) Cres and rough ** If he wo disurbances ravel in differen direcions hen when hey separae, each moves on as if nohing has occurred. c. - pulse P is ravelling along he rope O P O wih O being fixed. We can imagine ha here is anoher rope O which is he mirror image of O and he (a) P O P' ' refleced pulse is coming from O. he ime he pulse P reaches he reflecing surface, he image P reaches he inerface simulaneously. The displacemen of he poin O is he sum of he individual P' (b) (c) O P' P ' '

5 L Physics/Wave Phenomena/P.5 displacemen of P and P a O. However, as O is fixed, displacemen a O mus be equal o zero. Hence, he displacemen of P mus be negaive of ha of P or else he sum of he wo pulses could no be zero. Therefore, here mus be a phase difference π beween he inciden pulse and he refleced pulse. -If O is no fixed, i.e. a free end, he consrain on he displacemen of he paricle O will be removed and no phase inversion will occur. d. - When a ransverse wave on a spring is refleced a a denser medium (e.g. a fixed end or a heavier spring) here is a phase change of 80º (or π rad or λ/2).! Heavy spring Ligh spring Heavy spring Ligh spring " - Phase changes also occur when longiudinal waves are refleced. a fixed end a compression is refleced as a compression, a a free end a compression is refleced as a rarefacion. ξ R C x ξ C R x - When lighs his a medium of higher refracive index, i is refleced wih a phase change of π; when i his a medium of lower refracive index, i is refleced wih no phase change. C. Refracion see law of refracion in Huygen s Principle D. Polarizaion. Polarizaion of Waves - ransverse wave due o vibraions in one plane is said o be plane-polarized. The figure shows a plane-polarized wave due o vibraions in he verical plane yox (ypolarized) and anoher plane-polarized wave due o vibraions in he perpendicular plane zox (z-polarized).

6 L Physics/Wave Phenomena/P.6 - Longiudinal waves such as sound canno be polarized. y O x z 2. - Consider a horizonal rope D aached o a fixed poin D a one end. Unpolarized ransverse waves due o vibraions in many differen planes can be se up along by holding he end in he hand and moving i up and down in all direcions perpendicular o D, as illusraed by he arrows in he plane X. - Suppose here are wo parallel slis B and C beween and D as shown. - wave hen emerges along BC, bu unlike he waves along he par B of he rope (no shown), which are due o vibraions in many differen planes, he wave along BC is due only o vibraions parallel o he sli B. This plane-polarized wave passes hrough he parallel sli C. - Bu when C is urned so ha i is perpendicular o B, as shown in figure (ii), no wave is now obained beyond C. X polarized wave polarized wave (i) B C D X polarized wave (ii) B C D 3. Polarizaion of microwaves - grille of parallel meal rods is roaed beween (a) a source T of 3 cm plane-polarized E.M. wave (i.e. microwave) as shown; (b) deecor, a probe, wih a meer conneced o i. - When he rods are horizonal he meer reading is high, figure (i). So a wave ravels pas he grille. - When he grille is urned round so ha he rods are verical, here is no deflecion in he meer, figure (ii). Thus he wave does no ravel pas he grille. - If he elecric field is along he meal rods, he elecrons in he rods will be driven ino moion. Energy is ransferred from he microwave o he elecrons. The microwave is absorbed and canno pass on. If he elecric field is perpendicular o he rods, he elecrons canno go ino a corresponding moion. The microwave is no absorbed; i propagaes hrough he comb.

7 L Physics/Wave Phenomena/P.7 T (i) wave E meal rods wave E P high reading wave E 90º roaion (ii) P no wave zero 4. Polarizaion of ligh - Ordinary ligh from a lamp or he sun is unpolarized. - Producing polarized ligh:- a. Polarizaion by Selecive absorpion (by Polaroid) Polaroid conains long chain-like molecules, arranged o lie parallel o each oher. Elecrons can move freely along hese chains, bu no perpendicular o he chains. Ligh polarized perpendicular o he chains will pass hrough while ha polarized parallel o he chains will be absorbed. (The effec is like he meal grid in polarizaion of microwaves.) Example:- (i) mpliude of he Elecric field E afer passing hrough he polarizer =?? β β! E = E 0 cos β planepolarized ligh! E 0 polarizer! E 0 β E 0 sin β E 0 cos β ** In fac wha our eyes deec is he ligh inensiy, which is he energy per uni area per uni ime, bu no he ampliude of he elecric field. However, he inensiy of a propagaion wave is direcly proporional o he square of he ampliude of he 2 wave. i.e. I 0

8 L Physics/Wave Phenomena/P.8 (ii) Inensiy of he ligh afer passing hrough he polariser =?? β I =?? plane-polarized ligh wih inensiy I 0 polarizer mpliude of he inciden plane-polarised ligh E 0 = I 0 (Le he proporional consan be ) afer passing hrough he polariser, he ampliude of he ligh wave = E 0 cos β = I 0 cos β hence inensiy of he ligh observed = square of he ampliude = I 0 cos 2 β (iii) I =??? β I I 0 B - afer passing hrough he polariser, he unpolarised inciden ligh becomes plane polarised wih ligh inensiy I 0 /2 (and he ampliude of he wave = (I 0 /2), le he proporional consan = ) - passing hrough he polariser B, ampliude of he E-field becomes (I 0 /2) cos β, hence he inensiy of he ligh observed = (I 0 /2) cos 2 β b. Polarizaion by reflecion (i) - Looking a he reflecion of a lamp (unpolarised) from a glass plae hrough a Polaroid. - he observed inensiy changes when he Polaroid is roaed. - imply ha he refleced ligh is parially polarized, i.e. here is more linear polarizaion of one kind and less of he oher. -why?? (ii) - mechanism of producing refleced waves:- when he ligh ray passes ino he glass, elecrons in he glass are se ino vibraions along he elecric field, i.e. along he direcion ligh source * e - glass plae polaroid

9 L Physics/Wave Phenomena/P.9 of polarizaion. The oscillaing elecrons radiae E.M. waves, which form he refleced waves. ** Noice ha here is no waves generaed in he direcion of oscillaion and he ampliude of waves generaed is small for hose propagaing close o he direcion of oscillaion. (iii) - for polarizaion which is perpendicular o he plane of paper, elecrons vibrae in and ou of he paper, and radiae efficienly in he refleced direcion so refleced inensiy is srong. - for polarizaion which is in he plane of paper, he polarizaion of he ransmied i r srong wave is parallel o he direcion of vibraion of he elecrons. The refleced direcion is (a) close o he vibraion direcion of elecron, i r weak so he refleced inensiy is relaively low. θ - Hence he refleced ligh is preferenially polarized perpendicular o he plane of reflecion. - For he case ha he refleced ray is perpendicular o he ransmied ray, he θ (b) refleced ray becomes compleely polarized (perpendicular o he plane of paper). The inciden angle a which his occurs is called Brewser s angle. - Example: find he Brewser s angle for glass, given ha n =.5 inciden refleced θ b By Snell s law n a sin θ b = n g sin r n a = = n g sin (90º - θ b ) n g = n g cos θ b r Hence an θ b = n g /n a =.5 ransmied θ b = 56º c. Polarizaion by scaering (i) The ligh ray from he scaering of sunligh by air is parially polarized. Why?? (ii) - a ransparen ank wih cloudy waer (make by adding a drop of milk). - illuminae he ank by a narrow beam of unpolarised ligh. polaroid - direcion O: he direcion of observaion O is parallel o he vibraion along he x-axis, hence only he vibraion along he y- axis can be observed. The ligh is y-polarized. scaered ligh

10 L Physics/Wave Phenomena/P.0 direcion OB: he direcion of observaion OB is parallel o he vibraion along he y-axis, hence only he vibraion along he x-axis can be observed. The ligh is x-polarized. direcion OC: unpolarised. direcion OD: parially polarized, preferenially polarized in he horizonal direcion. D O C 5. pplicaion a. Reducing glare The Sun s rays refleced from glass, waer or scaered by he sky are preferenially polarized in he horizonal direcion. By making use of suiably oriened Polaroid discs, he glare can be reduced. B b. VHF and UHF anennas - FM radio broadcas in he VHF band makes use of horizonal polarizaion. FM anennas should be horizonal. - Television broadcas in he UHF band may eiher be on horizonal or verical polarizaion. The anennas should be oriened according o direcion of polarizaion of he received wave. E. Superposiion See principle of superposiion in phase inversion of reflecion. F. Beas - When wo sound waves of slighly differen frequencies bu similar ampliude are sounded ogeher, he loudness increases and decreases periodically and beas are said o be heard. - By principle of superposiion, he displacemen of he resulan wave is he sum of displacemen of individual wave. - an insan or C, he waves from he sources arrive in phase and reinforce o produce a loud sound. - an insan B, he waves from he sources are 80º ou of phase. The wo waves cancel each oher and lile sound is heard. - Bea frequency: Le T be he bea period (i.e. he ime beween wo successive maxima). Wihin T, one wave-rain of frequency f makes one cycle more han he oher frequency f 2. i.e. f T - f 2 T = f - f 2 = /T hence, bea frequency = f - f 2 = f

11 L Physics/Wave Phenomena/P. - use: Tuning of musical insrumen such as violin. Displacemen f f 2 (a) 0 Time Resulan Displacemen B C Variaion of ampliude (b) 0 Bea period = T Hence, Bea frequency = / T = f - f 2 Time G. Diffracion - The spreading of waves when hey pass hrough an opening or round an obsacle is called diffracion. - Diffracion is noiceable when he widh, a, of he opening is comparable o he wavelengh of he waves (i.e. λ a) or very small when he widh is large compared o he wavelengh. - Sound: long wavelenghs, hence can diffrac afer passing hrough doorways. visible ligh : shor wavelengh ( m), hence diffrac appreciably only hrough very small opening. - Parallel plane wavefrons are diffraced a a recangle sli B of widh a. The ligh passing hrough he sli is received on a screen S. lernae brigh and dark fringes are observed. Mos of ligh is concenraed in he cenral brigh band. a θ B some poins, he inensiy diminishes o a minimum (e.g. Q, Q 2,...) Posiion of he minimum is given by a sin θ = m λ (where m =, 2, 3,...) Q Q 2 S cenral brigh band O brighness dark λ θ = a λ θ 2 = 2 a

12 L Physics/Wave Phenomena/P.2 The angular widh of he cenral brigh fringe (which reflecs how serious he spreading of he waves is) is 2θ, where θ is he angle beween he direcion of maximum inensiy and he direcion of firs minimum inensiy. The angle is given by a sin θ = m λ wih m =. i.e. sin θ = λ/a When he sli is wide, a becomes large compared wih λ, hen sin θ is very small and hence θ is very small, no spreading occurs. When he sli is small, a becomes small compared wih λ, sin θ is very large and hence θ is very large and spreading (diffracion) is appreciable. H. Inerference. Inerference:- in a region where wave-rains from coheren sources cross, superposiion occurs giving reinforcemen of he waves a some poins and cancellaion a ohers Reinforcemen Cancellaion ** Coheren Sources:- sources have a consan phase difference, which means hey mus have he same frequency. The coheren sources required in Inerference can be obained by:- () division of wavefron, e.g. Young s double sli, Fresnel s biprism, Lloyd s mirror, ec. (2) division of ampliude, e.g. wedge fringes, Newon s rings, ec. 2. Division of wavefron a. Young s double sli experimen Principle:- Diffraced beam from S - monochromaic ligh (i.e. of one S color) from a narrow verical sli S S falls on wo oher narrow slis Monochromaic ligh source S S and S S and S 2 ac as wo coheren sources (boh being derived from (narrow) Single sli Double Diffraced beam sli from S 2 S, by division of wavefron). - diffracion causes he emerging beams o spread ino he region beyond he slis. - superposiion occurs in he shaded area, where he diffraced beams overlap and inerfere. inerference effecs in region where beams overlap

13 L Physics/Wave Phenomena/P.3 - alernae brigh and dark equally spaced inerference fringes can be observed on he screen. Theory:- - The pah difference beween waves reaching O from S and S 2 is zero, i.e. S O = S 2 O, hence, hey arrive in phase, i is a brigh fringe a O. - any poin P on he screen, he pah difference beween waves reaching from S and S 2 is S 2 which is approximaely given by S 2 = d sin θ for small angle θ, sin θ ~ an θ = x/l, hence he pah difference S 2 = dx/l maxima: There will be a brigh fringe if he pah difference is a whole number of wavelengh i.e. S 2 = nλ = d sin θ d sin θ = n λ (where n = 0,, 2,...) minima: There will be a dark fringe if he pah difference is (n + ½) λ i.e. d sin θ = (n + ½) λ (where n = 0,, 2,...) - Separaion of he fringes:- posiion of maxima is given by d sin θ = n λ or d x/l = n λ hence separaion x beween he nh and (n + )h brigh fringes is given d ( x/l) = λ i.e. x = λ L /d S S 2 d P θ x θ O L ppearance of Young s inerference fringes:- (i) x /d if λ and L consan (herefore he sli separaion should be small in order o increase he separaion); x L if λ and d consan (herefore he fringes should be viewed from a disance); x λ if L and d consan. (ii) If he source sli S is moved nearer he double slis he separaion of he fringes is unaffeced bu heir brighness increases. (iii) If he source sli S is widened he fringes gradually disappear. The sli S is hen equivalen o a large number of narrow slis, each producing is own fringe sysem a differen places. The brigh and dark fringes of differen sysems herefore overlap, giving rise o uniform illuminaion. I can be shown ha, o produce inerference fringes which are recognizable, he sli widh of S mus be less han λd /d, where D is he disance of S from he wo slis S, S 2. (iv) If one of he slis, S or S 2, is covered up, he fringes disappeared.

14 L Physics/Wave Phenomena/P.4 (v) (vi) If whie ligh is used he cenral fringe is whie, and he fringes eiher side are colored. Blue is he color nearer o he cenral fringe and red is farher away. The pah difference o a poin O on he perpendicular bisecor of he wo slis S, S 2 is zero for all colors, and consequenly each color produces a brigh fringe here. s hey overlap, a whie fringe is formed. Farher away from O, in a direcion parallel o he slis, he shores visible wavelengh, blue, produce a brigh fringe firs. The number of fringes obained depends on he amoun of diffracion occurring a he slis and his in urn depends on heir widh. The narrower he slis, he greaer will be he number of fringes due o he increased diffracion bu he fainer hey will be, since less ligh ges hrough. Necessary condiions for observable inerference:- (i) The wo waves mus have he same polarizaion. (ii) The wo waves mus have a consan phase difference. If he phase difference changes coninuously, hen here is someimes consrucive inerference and someime desrucive inerference. The average would be no inerference a all. coheren lengh wave rains If we use wo differen sodium lamps o illuminae he wo sli, inerference will no be seen. The reason is ha a sodium lamp emis ligh no coninuously bu in burss, which we call wave packes or wave rains. There is no definie phase difference beween he wave rains from he firs lamp and hose from he second lamp. Hence inerference is no observed. (iii) The wo waves mus have exacly he same frequency. (iv) The pah difference mus no be oo large. Consider wo waves from he same source, bu raveling over slighly differen disances o mee a a cerain poin. If he pah difference is larger han he lengh (coheren lengh) of wave rain, he received waves will come from differen wave rains, which are unrelaed and canno be coheren. ** The coheren lengh can be several meers for a laser, bu less han mm for a convenional ligh source. Examples:- S and S 2 are wo coheren ligh sources in a Young s wo-sli experimen separaed by a disance 0.50 mm and O is a poin equidisan from S and S 2. O is on a screen which is 0.80 m from he slis. When a hin parallel-sided piece of glass G of hickness is m is placed near S as shown, he cener of he fringe sysem moves from O o a poin P. Calculae OP if he

15 L Physics/Wave Phenomena/P.5 wavelengh of he monochromaic ligh from he wo slis is m and he refracive index of he glass is.5. S G P d = 0.50 mm θ L = 0.80 m O S 2 dsinθ Soluion: -If P is he cener of he fringe sysem, he number of waves is S P = he number of waves in S 2 P. - Since ligh ravels slower in he glass G han in air, he number of wavelenghs in G is more han in air of he same hickness as G. - The Exra pah difference due o he insering of G = n - = (n-) ** - This exra pah difference should be compensaed by he exra lengh of S 2 P, i.e. dsinθ (= d OP/L) OP - hence, d = ( n ) L ( n ) OP = L = 2.88 mm d ** No. of waves in air of hickness = /λ No. of waves in G of hickness = /λ = n/λ (i.e. = no. of waves in air of hickness n) hence, a glass of hickness can be viewed as air of hickness n. Variaion of inensiy of he fringe paern:- I/I min max max min 2 y/s = 0 consrucive inerference desrucive inerference - Le he inensiy due o each individual source be I 0. - hence ampliude of he wave from each individual source = I 0 - for brigh fringes: consrucive inerference, he ampliude of he resulan wave = 2 I 0 hence, inensiy of he brigh fringes = (2 I 0 ) 2 = 4 I 0

16 L Physics/Wave Phenomena/P.6 - for dark fringes: desrucive inerference, he ampliude of he resulan wave = 0 hence, inensiy of he dark fringes = 0. - If he wo sources are independen, no inerference paern will be observed. On he screen he ligh inensiy is uniform and is equal o 2I 0. b. Lloyd s mirror - plane mirror is illuminaed a almos grazing incidence by ligh from a sli S, parallel o he mirror. - virual image S 2 of S is formed close o S by reflecion and hese wo ac as coheren sources. - In he shaded area direc waves from S cross Monochromaic source * S Single sli Cross-wire of eyepiece or screen P C refleced waves which appear o come from S 2 and inerference fringes can be seen. - The expression giving he fringe spacing is he same as for he Young s double sli experimen. S 2 Glass plae - However, in Lloyd s mirror, if he poin P, for example, is such ha he pah difference S 2 P - S P is a whole number of wavelenghs, he fringe a P is dark, no brigh. This is due o he 80º phase change which occurs when ligh is refleced from he mirror. Ligh from S should fall a grazing incidence on he glass plae; he incidence is exaggeraed here c. Fresnel s biprism - Monochromaic ligh from a narrow sli S Single falls on a double glass prisms arranged as sli Biprism shown. - Two virual images S and S 2 are formed of S, one by refracion a each half of he prism, and hese ac as coheren sources. a S 2 S * S - n inerference paern is obained by he shaded region where he wo refraced beams overlap. Monochromaic source d - The heory and he expression for he fringe spacing are he same as for Young s mehod. Inerference effecs Cross-wire of eyepiece or screen 3. Division of mpliude a. Wedge Fringes - n air wedge can be formed from wo microscope slides clamped a one end and separaed by a hin piece of paper a he oher end. - monochromaic ligh from an exended source is parially refleced verically downwards by he glass plae G. When he microscope is focused on he wedge, brigh and dark equally spaced fringes are seen, parallel o he edge of conac of he wedge.

17 L Physics/Wave Phenomena/P.7 - Theory:- Some of he ligh falling on he wedge is refleced (i) upwards from he boom surface of he op slide and (ii) some of he res, (which is ransmied hrough he air wedge) is refleced upwards from he op surface of he boom slide. If l is he hickness of he air wedge a P hen he pah difference beween he rays a P = 2l. However, owing o he 80º phase change of he wave refleced a he op surface of he boom slide, here is an exra pah difference of half a wavelengh. Hence, he oal pah difference beween he wo waverains a P = 2l + ½λ for consrucive inerference: o.p.d. = nλ i.e. 2l + ½λ = nλ O θ 2l = (n - ½)λ (brigh fringe) for desrucive inerference: o.p.d. = (n + ½)λ i.e. 2l + ½λ = (n + ½)λ 2l = nλ (dark fringe) separaion of he fringes: 2l n = nλ 2l n+ = (n + )λ hence, 2(l n+ - l n ) = λ 2 l = λ for small angle θ, θ = l/ x hence, l = θ x hence, 2 θ x = λ x = λ/(2θ) ** Separaion x increases, as λ increases. Separaion x increases, as θ decreases. Travelling Microscope G ir wedge Monochromaic ligh Q P l l x Sodium lamp Glass plaes - Quesion: Wha would happen if (i) waer is inroduced ino he region beween he slides? (ii) he conac side of he upper slide is uplifed slowly? b. Newon s rings - Monochromaic ligh (e.g. from a sodium lamp) is refleced by he glass plae G so i falls normally on he air film formed beween he convex lens of long focal lengh and he fla glass plae. - Inerference occurs beween ligh refleced from (i) he lower surface BC of he lens and

18 L Physics/Wave Phenomena/P.8 (ii) he upper surface DBE of he plae. - Opical pah difference beween he wo ray a P = 2PQ + ½λ (he exra o.p.d. ½λ is due o he phase inversion when ligh is refleced from he upper surface DBE of he plae.) - Consrucive inerference: if he o.p.d. = mλ (brigh ring) Desrucive inerference: if he o.p.d. = (m + ½)λ (dark ring) - series of brigh and dark rings is observed. - The rings are fringes of equal hickness. - s he radii of he rings increase, he separaion decreases, why? - he cener B of he fringe sysem where he geomerical pah difference beween he wo waves is zero, here is a dark spo, why? - In wha way would he Newon s rings paern change if he convex lens is moved upwards slowly? Plano-convex lens D Fla glass plae S r n B L O B R r n Travelling microscope G C ir film E C P l Q Sodium lamp 4. Using inerference a. Tesing of opical surfaces - Fringes of equal hickness are useful for esing opical componens. - Example: (i) In he making of opical flas, he plae under es is made o form air wedge wih a sandard plane glass surface. ny uneven pars of he surface will show up as irregulariies in wha should be a parallel, equally spaced sraigh se of fringes. (ii) The grinding of a lens surface may be checked if i is placed on an opical fla and Newon s rings observed in monochromaic ligh. The rings should be exacly circular if he lens is spherical. b. Non-reflecing glass (Blooming of lenses) - In opical insrumens conaining lenses or prisms ligh is los by reflecion a each refracing surface and resuls in reduced brighness of he final image. - The amoun of ligh refleced a a surface can be reduced by coaing i wih a film of ransparen maerial (e.g. magnesium fluoride) o a hickness of ¼λ of ligh in he film. - Ligh refleced from he op (ray ) and boom (ray 2) surfaces of he film inerfere. - as he o.p.d. beween he wo rays = 2l = 2(¼λ ) = ½λ l!" ir Film Glass

19 L Physics/Wave Phenomena/P.9 ( l = ¼λ = λ,where λ is he wavelengh of ligh in he film, 4 n λ is he wavelengh of ligh in air, n is he refracive index of he maerial of he film.) ** There is no need o consider he phase inversion. The refracive index of he film is less han ha of glass and so each reflecion (boh he op and boom reflecions) suffers a 80º phase change. The ne effec of he phase changes on he pah difference is hus zero. - The wo rays inerfere desrucively, i.e. here is no refleced ligh. - The inerference is complee for one wavelengh only, usually aken o be ha a he cener of he visible range. For red and blue ligh he reflecion is weakened bu no eliminaed and a coaed or bloomed lens appears purple in whie ligh. - e.g. λ = 550 nm and n =.38, hen l = 00 nm. 5. Everyday examples of inerference a. Colors of oil films on waer - Inerference occurs beween wo wave-rains: (i) one refleced from he surface of he oil and ir (ii) he oher from he oil-waer inerface. Oil - When he pah difference gives consrucive Waer inerference for ligh of one wavelengh, he corresponding colour is seen in he film. - s he pah difference varies wih he hickness of he film and angle of viewing, difference colour is seen a differen pars of he film. b. Verical Soap film (i) a verical soap film illuminaed by monochromaic ligh: - firs he film appears uniformly coloured. - s he soap drains o he boom, a wedge-shaped film of liquid forms, he op of he film being hinner han he boom. Horizonal brigh and dark fringes are observed across he film. - When he upper par of he film becomes exremely hin (small han λ /4), a black fringe is observed a he op (he black fringe is due o he 80º phase change by reflecion). nd he film breaks shorly aferwards. (ii) a verical soap film illuminaed by whie ligh: - a succession of broad coloured fringes, viole o red, is firs observed. - The fringes widen as he film drains, and jus before i breaks a black fringe is obained a he op. * Soap film

20 L Physics/Wave Phenomena/P.20 c. Pulsing of he picure on a elevision receiver - When an aircraf passes low overhead, he signal ravelling direcly from he ransmiing o he receiving aerial inerferes wih ha refleced from he aircraf. 6. Diffracion graing a. diffracion graing is a large number of close parallel equidisan slis, ruled on glass or meal (e.g. 300 lines per mm). b. Theory: - a plane wave of monochromaic ligh of wavelengh λ falls on a ransmission graing in which he sli separaion (called he graing spacing) is d. - consider he wave rains coming form he slis and ravelling a an angle θ o he direcion of he inciden beam. - The pah difference is dsinθ for all pair of waves from any wo successive slis. -If d sinθ = nλ, where n is an ineger, hen reinforcemen of he wave rains occurs in direcion θ and a maximum will be obained. - Noice ha he condiion for maximum in diffracion graing is he same as ha in Young s double sli experimen. - Example: a diffracion graing of 300 lines per mm, ligh of wavelengh λ = m. Wha is he maximum order of brigh fringe ha can be viewed? graing spacing d = 0-3 /300 maximum possible opical pah difference is d. d/λ = 5.5, hence he maximum order ha can be viewed is 5. Inciden beam - λ d Graing I / I 0 -/2 0 I / I 0 d sin θ x θ /2 n = 3 θ θ 2 n = /3 -/3 0 /3 2/3 θ 3 n = 2 n = 2 N = 2 N = 3 n = n = 0 n = y/s y/s c. Variaion of inensiy - For a diffracion graing, he brigh fringe is narrow and he inensiy is very high (compared wih ha in Young s double slis experimen). I / I 0 x N = 5-0 y/s

21 L Physics/Wave Phenomena/P.2 - In fac, he widh of he brigh fringe, x, is inversely proporional o he number of slis N. i.e. x /N. he inensiy of he brigh fringe I is direcly proporional o he square of he number of slis N. i.e. I N². d. pplicaion (i) Deerminaion of wavelengh using a diffracion graing - because he inerference maxima are much sharper and brigher, a diffracion graing permis a much beer deerminaion of wavelengh. - Pu a sodium lamp (wih a verical sli) on he bench. Place a meer rule poining a he P ligh source abou 3 m away, and anoher mere rule B perpendicularly a one end of. Place a fine diffracion graing wih he sodium lamp B mere rule θ graing slis verical. View he illuminaed sli (a) hrough he graing and mark he angular diffracion posiions of he images by moving a pencil P graing along B. - Since he ligh source is sufficienly far away from he graing, which is perpendicular o he direcion of inciden ligh, he siuaion is one of normal incidence. - The corresponding disance of he pencil from is y m. Then he angular posiion of he mh order maximum is given by P y m an θ = y m and he wavelengh λ of he ligh can be calculaed by d sin θ = m λ (b) m θ θ provided ha he graing spacing d is known. (ii) Specra nalysis - If whie ligh is inciden normally on a diffracion graing several colored specra are observed on eiher side of he normal. - ll he colors coincide in he cener, forming a whie line. - s viole has a shorer wavelengh han red, θ is less for viole han for red. Consequenly he specrum colors on eiher side of he inciden whie ligh are viole o red. - The second order band is wider han he firs order diffracion band; he hird order diffracion band is wider han he second diffracion band, and so on (i.e. W V R V 2 V 3 R 2 R

22 L Physics/Wave Phenomena/P.22 he dispersion increases wih he order). Therefore a high resoluion of he specrum can be obained if a high order paern is observed. (Remark: The dispersion is also inversely proporional o he graing spacing a. Hence in order o have high resoluion, a high order paern should be observed hrough a fine graing wih small graing spacing a.) red 0 0.7/a.4/a 2./a /a 0.8/a.2/a.6/a 0 viole - The second and hird order bands overlaps; he hird and fourh order bands overlap, ec. The condiion for wo colors in wo differen bands o overlap is given by sin θ sin θ a sin θ = m λ = m 2 λ 2 where m, m 2 are order numbers and λ, λ 2 are he wavelenghs of he wo colors concerned. m = 2, m 2 = 3 and he wavelengh for red, λ = 700 nm. By he condiion above, he minimum wavelenghs of he ligh ha will overlap wih he red ligh is 467 nm. s wavelengh for viole is abou 400 nm which is shorer han 467 nm, he second and he hird mus overlap. - Condiions for a good diffracion graing: (i) small graing spacing a, (high dispersion). (ii) large oal number of sli, N (sharper and brigher line).

23 L Physics/Wave Phenomena/P.23 ** Supplemenary Noes for diffracion. Suppose parallel plane wavefrons from a disan objec are diffraced a a recangular sli B of widh a. 2. Consider a plane wavefron which reaches he opening B. ll poins on i beween and B are inphase, ha is, hey are coheren. Their combined effec a any disan poin can be found by summing he numerous waves arriving here, from he Principle of Superposiion. 3. Cenral brigh image - Consider a poin O on he screen which lies on he normal o he sli passing hrough is midpoin C. O is he cener of he diffracion image. I corresponding o a direcion θ = 0. - The waves from he secondary sources such as, X, Y, B have no pah difference. - ll he waves arrive in phase a O. The cener of he diffracion image is herefore brighes. - In a direcion very slighly inclined o θ = 0, he waves from all he sources beween and B arrive slighly ou of phase a he corresponding poin P of he image near he cener. So he brighness decreases. 4. Firs minimum - In a paricular direcion θ, we reach he dark edge Q of he cenral image. This direcion corresponds o a pah difference of λ beween he wo sources a he edges and B of he sli. - Divide he wavefron B ino wo halves. - The op poin of he upper half C, and he op poin C of he lower half BC, send ou waves o Q which have a pah difference λ/2, from above. Thus he resulan ampliude a Q is zero. - ll oher pairs of corresponding poins in he wo halves of B, for example, X and Y where CX = BY and he wo boom poins C and B, also have a pah difference λ/2. So he brighness a Q is zero. - Condiion for he firs minimum is d sin θ = λ.

24 L Physics/Wave Phenomena/P Secondary fringes - Secondary brigh are also obained on he screen beyond Q. - Poin T which lies in a direcion θ 3 where he pah difference of waves saring from and B is 3λ/2. - The wavefron B divided ino hree equal pars. The waves from he exreme ends of he upper wo pars have a pah difference λ. Thus, hese wo pars of he wavefron produce darkness a T. The hird par produces a fringe of ligh a T much less brigh han he cenral fringe. 6. Minimum brighness of he secondary fringes - The direcions of he minimum brighness of he secondary brigh fringes can be found by dividing he diffraced wavefron a he sli ino four (or six, and so on) equal pars in he same way as he cenral brigh fringe. - The direcions for he successive minima are given by d sin θ = m λ (where m =, 2, 3,...)