sin r sini c v c c ALFA PHYSICS CLASSES RAY OPTICS

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1 RAY OPTICS Ray : The straight line path along whih the light travels in a homogeneous medium is alled a ray. The arrow head on the ray gives us the diretion o light and number o rays ombined together is alled beam o light. Reration : The phenomenon in whih ray o light traveling rom one medium to another medium o dierent optial density, deviates rom its original straight line path is alled reration o light. When light moves rom rarer to denser medium it bends towards the normal and when it moves rom denser to rarer medium it bends away rom the normal. Reration o light ours beause the speed o light hanges as one moves rom one medium to another. Also, the wavelength o light hanges, but requeny and phase o the wave remains onstant on reration I,e, ni hange in phase or requeny ours. Laws o Reration: First Law [Snell s Law] The ratio o the sine o angle o inidene to the sine o angle o reration is onstant or a pair o media in ontat. This onstant is equal to the rerative index o seond medium w.r.t. irst medium. The irst medium is one in whih inident ray lies and the seond medium is one in whih the rerative ray lie. I and denotes the rerative index or the two mediums then sini sin r Seond law: the inident ray, rerated ray and normal all three lie in the same plane whih is plane perpendiular to the rerating surae. Rerative index o the medium an also be explained in terms o the veloity o light in any given medium. Absolute rerative index o the medium is the ratio o veloity o light in vauum to the veloity o light in that medium medium Also, Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar air v

2 v When the ray o light moves rom denser This implies that the veloity o light dereases i the medium hanges times. T Real and Apparent epth : Whenever an objet is plaed in optially denser medium, like objet O plaed at the bottom o the ontainer, the ray o light starting rom objet moves rom denser to rarer medium and bends away rom normal. Thus a virtual image o the objet is ormed at I. Then, distane OA is alled real depth and IA is alled apparent depth o objet. Now, Using Snell s law, AB sin i and sin r OB sin i sin r IB OB I rarer medium is air, then and I angles are small then OB OA and IB IA AB IB OA IA OB IB Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar Realdepth Apparent depth The shit in the position whih takes plae ater reration rom the surae is is x OA-IA here h denotes the real depth o the objet. Reration through Compound Plate: OA x OA h x h h Consider a ompound plate made o two materials with rerative index b and ( > b). A ray o light inident on ray moving rom rarer to denser medium bends towards the normal. Using Snell s law, sin i b sin r a

3 Similarly, at ae M M it suers reration and using Snell s law, sin r sin r b Finally at surae M M it suers reration and omes out parallel to inident ray as all the rerating suraes are parallel. Multiply, all three equations, sin r sin r a Total Internal Reletion: a a b b b b This phenomenon is observed when a ray o light moves rom denser to rarer medium. When the angle o inidene in suh a ase is greater than the ritial angle then light would be releted bak into the same medium and phenomenon is alled total internal reletion. a a a Consider a soure o light S situated in denser medium say water. As the rays move rom denser to rarer medium they bends away rom the normal. I we go on inreasing the angle o inidene angle o reration also goes on inreasing (aording to Snell s law). At one partiular angle o inidene, angle o reration beomes 90º. The angle o inidene or whih the angle o reration is 90º is alled ritial angle. I angle o inidene is inreased urther the ray gets totally releted bak into the same medium instead o reration. At ritial angle, i, r 90. sin i sin 90 sin i Appliations o Total Internal Reletion:. Mirage Formation: It is an optial illusion whih takes plae in hot ountries. The layers o earth in ontat with the earth are hooter and rarer whereas the upper layers are older and denser. When the ray o light moves Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

4 . downwards ater reletion rom objet like tree it is moving rom denser to rarer medium. The agle o inidene goes on inreasing with reration are eah layer o atmosphere. For one partiular layer the angle o inidene is greater than ritial angle and the ray o light suers total internal reletion bak in the upward diretion. Thus a virtual and inverted image o the objet is ormed on the ground. These virtual images produes the impression o reletion rom water due to atmospheri disturbane. 3. Optial Fibres : Optial ibres onsists o several thousand o very long ibres o the diameter o 0 4 m, with rerative index.7. The ibres are loated with a thin layer o material o lower rerative index. Light entering rom one side undergoes about 0 - thousand reletions per meter and omes out rom other end. Optial ibres an be put to number o appliation ; (i) (ii) (iii) They an be used to transmit high intensity laser light insider the body. They an be used in the ield o ommuniation in sending video signals rom one plae to another. They are used to see images o body parts not learly visible in X - Rays. Reration Through Spherial Suraes : A spherial surae is ormed i the rerating surae orms the part o a sphere. The surae is said to be onvex i it bulges towards the rarer medium side and it is onave surae i it bulges towards denser medium side. Sign Conventions : The ollowing sign onventions are used or reration at single surae.. All the distanes are measured rom pole o spherial suraes.. The ray o light moves rom let to right with pole - the origin o artesian oordinate system. The distanes to the right o the pole are positive and to the let o the pole are negative. Assumptions :. The objets are assumed to be point objets lying on the prinipal axis.. The aperture o spherial surae is small. 3. Inident and rerated rays makes small angles with prinipal axis. Ray o light Moving rom Rarer to enser Medium: a) With onvex towards rarer (Real Image): Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

5 Consider an objet O lying on the prinipal axis and the inident ray o light starting rom O makes an angle a with the prinipal axis. Let CA be the normal and inident ray makes angle i with it. As ray moves rom rarer to denser medium it bends towards the normal and the bending is just suiient to make the rerated ray meets the prinipal axis at I. The rerated ray makes angle a with the prinipal axis and r with the normal. Using Snell's law, I angle o inident and reration are small, then, sin i i i r sin i sin r and sin r r i r Also i + and r (beause exterior angles are equal to interior opposite angles) As angle, and are small, Substituting these values in (), we obtain, ( ) ( + ) ( ) + tan tan tan MP ' CP R AP' AP' OP' OP AP' AP' IP' IP AP' AP' CP' CP MP MP + OP IP + u v () (as aperture is small, OP' OP, AP' AP, CP' CP) Virtual Image: Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

6 In this ase inident ray OA moving rom rarer to denser bends towards the normal but the bending is not suiient to make it move towards prinipal axis. Thus, a virtual image o objet is ormed at I. Using Snell's law, sin i sin r I the angle o inidene and reration are small then sin i ~ i and sin r ~ r, i r i r Also, i + and r + (beause exterior angles are equal to interior opposite angles) ( + ) ( + ) ( ) () Substituting the values o,, in (), we obtain ( MP MP ' MP ) CP OP IP R u v R v u (b) With Conave Towards Rarer Medium: Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

7 The ray o light starting rom point objet O lying on the prinipal axis moves towards the normal as it moves rom rarer to denser medium and virtual image o the objet is ormed at I. Using Snell's law where sin i sin r I the angle o inidene and reration are small then sin i ~ i and sin r ~ r, i r i i and r ( ) ( + ) ( ) + () r Substituting the values o,, in (), we obtain ( MP MP ' MP ) + CP OP IP + R u v R v u Similarly, we an prove the idential results or light moving rom denser to rarer medium. Lens : A portion o rerating material bound between two spherial suraes is alled lens. (i) A lens is said to be onverging i the width o the beam dereases ater reration through it. Foal length o onverging lens is taken as positive. Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

8 (ii) A lens is said to diverging i the width o beam inreases ater reration through it. Foal length o diverging lens is negative. einitions Regarding Lenses : Optial Centre : It is a point lying on the prinipal axis o lens within or outside it, suh that ray o light passing through it goes undeviated. I the two suraes are o same radii o urvature then optial entre lies exatly in the entre o the lens. Radius o Curvature (R & R ) : part. Radius o urvature o a surae o lens is deined as the radius o that sphere o whih surae orms a Prinipal Axis : axis. The line joining entre o urvature o two suraes and passing through optial entre is alled prinipal Prinipal Fous : Prinipal ous o the lens is a point at whih beam o light oming parallel to the prinipal axis atually meets or appears to meet ater reration through lens. Foal Length : Foal lenght o a lens is deined as the distane between ous and optial entre. It is denoted by. Foal Planes : It is plane passing through the prinipal ous and perpendiular to prinipal axis. Len's Formula : Lens ormula is a relation between oal length o lens with the distane o objets and images. Convex lens: Let AB be the objet plaed on the prinipal axis and beyond ous F. The ray starting rom A passing through optial entre goes undeviated and the ray moving parallel to prinipal axis passes through ous. The two ray meet at A, then AB is the image o the objet AB. As ABC and ABC are similar, Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

9 AB A B BC B C...() Also, CF and ABF are similar, Also, C AB > From () and (), ividing by uv, AB A B C A B CF FB CF FB BC B C u v CF FB v u v uv v u...() Virtual Image: I the objet lies between optial entre and the prinipal ous then a virtual image o the objet is ormed. Again as ABC and ABC are similar. Similarly, as CF and ABF are similar rom () and () and C AB, AB A B AC B C C A B CF FB...()...() Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

10 AC A' C u v CF FB v + u v uv ividing by uv, v u Linear Magniiation: The linear magniiation produed by a lens is the ratio o size o the image to the size o the objet. For virtual image, size o image(a' B') A' C m size o objet (AB) AC h m h v u h m h v u ( or real image ) ( rom()) Thus, or a onvex lens, linear magniiation is positive when image is virtual and negative i image is real. Similarly, or onave lens the linear magniiation is always positive. h m h v u Lens Maker s ormula: Consider a thin lens with optial entre C, and the point objet O plaed on the prinipal axis o this lens. Light originating rom the objet on prinipal axis ater reration at the irst surae heads towards I. Reration takes plae at the seond surae and inal image is ormed at I. Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

11 Reration At APB : Consider reration at irst surae only, and are the angles whih the inident ray, rerated ray and normal to the irst surae makes with the prinipal axis. Using Snell's Law, sin i sin r sin i sin i As angle o inidene and reration are small, thereore, sin i i and sin r r Also i r i + and r ( + ) ( ) ( ) + () Reration at seond surae APB: I there is no seond surae the rerated ray rom irst surae meets the prinipal axis at I but in moving rom denser to rarer medium reration takes plae at seond surae. The ray bends away rom normal to seond surae and inal image is ormed at I. Using Snell s law, sin i sin r As angle o inidene and reration are small, thereore, sin i i and sin r r Also Adding () and (), we get I angle,, and are small, Substituting these values in (), we obtain, i r ( + ) ( + ) ( ) () + ( ) ( + ) MP tan OP T P tan I P MP tan C P T P tan C P (3) Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

12 + ( OC I C v u ) CC R R ( ) + R R Similarly, the relation an be proved or onave lens also. + CC Power o a Lens : Power o lens is the ability o lens to onverge or diverge a beam o light alling on the lens. Mathematially, it is equal to the reiproal o oal length. P ( ) R R Units o power is ioptres, i oal length is measured in meters. The number o lenses an be ombined to inrease the magniiation (ompound mirosope), make the inal image eret (terrestrial telesope). As eah lens has its own magniying power, the resultant magniiation is the produt o magniiation o individual lenses i.e. and resultant power is m m m..... m n P P + P P n I we have two lenses with distane d between them, then resultant power is P P + P dp P Reration through Prism: A prism is a wedge shaped body made rom rerating medium bound by two plane aes inlined to eah other at same angle. The two plane aes are rerating suraes and angle between them is the angle o prism. Consider ABC as the prism with AB and AC as the two rerating suraes. The inident ray PE meets the rerating ae AB at E making an angle o inidene i with normal NN. As it is moving rom rarer to denser medium it bends towards the normal making an angle r. Similarly, at seond ae it moves rom denser to rarer Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

13 medium making an angle o inidene r and angle o reration e (or angle o emergene). The angle between inident and rerated ray is alled angle o deviation. Again in quadrilateral AENF, EF + FE (i r ) + (e r ) (i + e) (r + r ) () AEN + AFN 80º so ENF + A 80º () Also in ENF, From () and (3) r + r + ENF 80º A r + r 3 (i + e) A For prism having small rerating angle A the inident ray makes small angle with prism, thus angle o reration is also small. Applying Snell s law, or reration at ae AB and AC, sin i sin r sin sin r (4) (3) I the angle o inidene and reration are small, then i r and e r (r + r ) A ( ) A Angle o Minimum eviation: The minimum deviation value o angle o deviation when ray o light passes through the prism is alled the angle o minimum deviation. In minimum deviation position, Also, Using Snell s law, m i A A i m + sin i sin r A sin sin + m A i e and r r A r + r r m i + e A (where r r r) An alternate proo will be done in the lass. Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

14 ispersion: A ray o light while passing through a prism spilts up into its onstituent wavelengths. The phenomenon is known as dispersion. As the deviation suered by violet and red light is dierent (Cauhy's relation), thereore, olours o dispersed beam will spread in a one o angle v r. This dierene o deviation produed in violet and red light is alled angular dispersion. v r [( v )A] [( v )A] ( v r)a () I deviation suered by mean light is, () and () > v ( ) A r ( r v ) ( ) () where is alled the dispersive power o prism. ispersive power is thus deined as the ratio o angular dispersion to mean deviation produed by prism. As v > r, thereore, dispersive power is always positive. eviation without ispersion: In this ase, the ray o white light entering the prism omes out as white light. The angles and material o prism are so adjusted that the dispersion produed in one prism gets exatly anelled out by the other. Suh a prism is also alled ahromati prism. Consider two prism one o rown glass and other o lint glass having angles o prism A and A respetively. The rerative indies be and respetively. ispersion produed in two prisms, ( v r) + ( v r) ( v r) A + ( v r) A 0 (v r) + (v r) 0 as A ', ' A ' + 0 whih is the neessary ondition or ahromatism. The net deviation produed in the prism is given by, ( ) A + ( ) A v r ( ) A + ( ' ) v r v r ( ) A + v r ( ) A A 0 A 0 ispersion without eviation: Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

15 In this ase, inident ray and emergent ray are parallel to eah other and emergent ray onsists o rays o various olour in dispersed orm. As net deviation produed is zero, + 0 ( )A + ( ) A 0 ( ) A A ( ) or A ( ) A Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

16 Spetrometer: Constrution : The basi parts o spetrometer are as ollows : Collimator : It is an arrangement o two tubes, one itted with slit and other with onvex lens. One tube an be slided into other using rak and pinion arrangement. Prism Table : It onsists o irular dis, and held together by means o three srews P, Q & R. Telesope : It onsists o two onvex lenses attahed to two ends o two tubes. One tube slides into the other using the rah and pinion arrangement. Sale : It onsists o main sale, irular in orm, attahed with two vernier systems V and V 80º apart. The sale is attahed to telesope and rotates along with it while the vernier systems are attahed to prism table. etermination o A : Plae the prism on the prism table in suh a way that its rerating edge aes the ollimator and light rom ollimator is inident on both the ous. Look through ae AB. An image o slit S, due to light releted rom AB is seen. Note the readings o both the verniers. Now bring the telesope towards other ae AC and loate the image o slit. Again note the readings o two verniers. The dierene between two readings be and angle o prism, A /. etermination o m : Plae the prism on the table suh that light is inident on one o the aes and omes out o the other. Bring the telesope in line with the emergent beam. Change the angle o inidene by rotating the prism table in suh a way that the image o slit moves towards the diret line. It will be observed that the image o slit omes to stand still at some position. On turning the table urther in same diretion the image moves bak. When the slit omes to rest, lok the table. The deviation produed in this ase is minimum. Substituting or A and m, m an be alulated. Sattering : Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

17 When a beam o white light is passed through a water tank ontaining ew drops o milk, the olor o light when observed at right angles to its propagation is rih in blue. This ould be explained due to sattering. When a beam o light is inident on partiles o very small size, smaller than the order o wavelength o light, light proeeds in all possible diretions. The phenomenon is alled sattering. Aording to Rayleigh's law " The intensity o sattered light, having wavelength l, varies inversely as ourth power o its wavelength." As r b, thereore, sattering o blue olour will be 6 times more than that o light. I 4 Blue olor o Sky : Being shorter wavelength, sattered blue light dominates and hene sky appears blue. Rainbows : The olorul display o onentri ars observed in the sky, as viewed by the observer with bak towards sun are alled rainbow. There are generally two rainbows primary and seondary. Primary rainbows has violet olor on the inner side and red on the outer, Seondary rainbow has red on the inner side and violet on the outer side. Some other bows in addition to these two are alled supernumerary bows. Their nature depends upon the size o water drops and are aused by diration phenomenon. Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

18 Primary Rainbow : It is ormed due to single total internal reletion rom water drop suspended in air. Consider a water drop having its entre at O. Let a ray SA be inident on the drop, at A with angle o inidene i and angle o reration r. At B the ray suers total internal reletion and inally enters the eye ater getting rerated at C. eviation at A i r, deviation at B ( r), deviation at C (i r). For deviation to be maximum or minimum, d / di 0, Aording to Snell s law, ierentiating w.r.t. i, As Total deviation + i 4r dr dr 4 0, di di sin i sin r, osi osr sin i sin r dr di dr, osi osr di osi osr Squaring 4os 4 4sin sin i i Angle subtended by emergent red ray 43 0 and or violet ray it is 4º. All rain drops lying on onentri ars having their entres on the line joining sun with observer eye show similar type o dispersion. Thereore, violet olor orms base o one as shown. os i 4 sin r r Formation o Seondary Rainbow : Seondary rainbow is ormed due to two total internal reletions o a ray o light within a water drop. Consider a ray SA inidene on a drop at A, at an angle i, the reration angle be r. The ray suers total internal reletions at B and C. Finally it enters the eye ater undergoing reration at. Net deviation, eviation at A i r eviation at B r eviation at C r eviation at i r Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

19 (i r) + ( r) + (i r) + i 6r For deviation to be minimum, d 0 di dr 6 0 di dr di 3 Proeeding as we did in primary rainbow, sin i Angle subtended by red ray with horizontal 5º Angle subtended by violet ray with horizontal 54º 9 8 All the drops situated on onentri ars having urves on the line joining sum with observers eye will behave in similar manner. That is why red appears on inner side and violet on outer side. Fraunhoer Lines (Solar Spetrum) : It has been observed that solar spetrum onsists dark lines in the solar spetra. These lines are alled Fraunhoer lines. Sun onsists o three layers with innermost layer alled photosphere with temperature o the order o 0 x 0 6 K. Seond layer is the sun's atmosphere whih onsists o gases mainly hydrogen and helium. The temperature is very high o the order o thousands o degree entigrade. The outermost part o the sun is hromosphere with temperature o about 6000ºC. Light emitted rom sun's atmosphere onsists o ontinuous spetra. As the light passes through the hromosphere, various elements present there absorb the wavelengths whih they themselves will emit when hot. This results in appearane o dark lines in the spetra. Fraunhoer lines are used to study elements present on sun. Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

20 Simple Mirosope:... 0 Magniying Power :... 0 Compound Mirosope:... Magniying Power:... Astronomial Telesope:... Normal Adjustment:... When image is at least distane o distint vision:... Magniying Power:... 3 Simple Mirosope: Objet is plaed between onvex lens and prinipal ous an eret, virtual and magniied image is ormed on the same side o the objet. In the igure objet AB whih when viewed by an unaided eye annot be seen distintly. A onvex lens is then interposed between the eye and the objet so that the distane 'a' o the objet rom the lens is less than the oal length o the lens. A virtual, eret and magniied image A'B' will be produed. By adjusting the distane o objet image is ormed at least distane o distint vision Magniying Power : It is the ratio o angle subtended by the image at the eye to the angle subtended by the objet at the eye when both are plaed at least distane o distint vision. Magniying Power AB CB A B' CB' CB' CB u tan tan Sine the virtual image is ormed at least distane o distint vision, thereore, V. Using Lens Formula, v u Multiplying both sides by, we get, u + u + u + Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar...()...()

21 From () and (), M + Compound Mirosope: It is used where larger magniiation is required. The onvex lens O o short oal length and short aperture and eye piee o short oal and large aperature is required. Let AB be an extended objet situated on the prinipal axis at distane greater than oal length o the objetive. As reration takes plae through the objetive O, a real inverted and magniied image A B is ormed. The position o eye piee so adjusted that A B alls within its oal length and so the inal image A B is ormed at least distane o distint vision. Thus, inal image A B is ormed in highly magniied but is inverted with respet to the objet AB. The ourse o rays orming the inal image. Magniying Power: The magniying power is deined as the angle subtended by the inal image at the eye to the angle subtended by objet when both are plaed at least distane o distint vision rom eye. For objetive lens, M tan M tan A'' B''' A' B' M O M C A' B' AB v M O u Again sine the lens E, ats like simple mirosope, its magniiation M C is given by, M C + Thus, magniiation o ompound mirosope should be, v M + u e e Astronomial Telesope: evie used to see very ar o heavenly bodies. The objetive lens has large oal length and large aperture. The eye piee has small oal length and small aperture. A parallel beam o light oming rom distane objet orms a real, inverted and diminished image at a distane 0 rom O. The image then ats as an objet or eye piee, and inal image is ormed ater reration through eye piee. Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

22 Normal Adjustment: I the inal image is ormed at ininity ater reration through the eye piee. The magniying power o telesope is deined as the ratio o angle b subtended by the image to the angle subtended by the objet at the eye when both are plaed at ininity. C B' C B' The distane between the two lenses is ( 0 + e). M 0 e M tan tan A' B' CB' CB' C B' A' B' C B' ( oal length o objetive ) ( oal length o eye piee) M When image is at least distane o distint vision: The objetive lens orms the real inverted and diminished image A B at n. I A B orms the real image within the oal length e o the eye piee, a inal virtual but magniied image A B is observed. The position o eye piee is so adjusted that inal image is ormed at least distane o distint vision rom the eye. 0 e Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar

23 Residene: 49, Chotti Baradari Part -[ near m edial College], G arha Road Jalandhar Magniying Power: It is the ratio o angle subtended at the eye by the inal image ormed at least distane o distint vision to the angle subtended by the unaided eye by the objet at ininity. u e B C B C A B B C B C A B M 0 ' ' ' ' ' ' ' ' tan tan Using Lens ormula, M u u u v e e e e e e 0