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1 . CONDITION FOR F RECTILINEAR PROP OPAGATION OF LIGHT : (ONLY FORF INFORMA ORMATION NOTE IN JEE SYLLABUS) Some part of the optics ca be uderstood if we assume that light travels i a straight lie ad it beds abruptly whe it suffers reflectio or refractio. The assumptio that the light travels i a straight lie is correct if (i) the medium is isotropic, i.e. its behavior is same i all directios ad (ii) the obstacle past which the light moves or the opeig through which the light moves is ot very small. Cosider a slit of width a through which moochromatic light rays pass ad strike a scree, placed at a distace D as show. It is foud that the light strikes i a bad of width b more tha a. This bedig is called diffractio. Light beds by (b-a)/ o each side of the cetral lie.it ca be show by wave theory of light that siθ a λ...(a), where θ is show i figure. This formula idicates that the bedig is cosiderable oly whe a ~ λ. Diffractio is more proouced i soud because its wavelegth is much more tha that of light ad it is of the order of the size of obstacles or apertures. Formula (A) gives b a λ. D a It is clear that the bedig is egligible if D λ << a or a a >> Dλ.If this coditio is fulfilled, light is said to move rectiliearly. I most of the situatios icludig geometrical optics the coditios are such that we ca safely assume that light moves i straight lie ad beds oly whe it gets reflected or refracted. Thus geometrical optics is a approximate treatmet i which the light waves ca be represeted by straight lies which are called rays. A ray of light is the straight lie path of trasfer of light eergy. Arrow represets the directio of propagatio of light. Figure shows a ray which idicates light is movig from A to B.. PROPERTIES OF LIGHT (i) (ii) (iii) (iv) GEOMETRICAL OPTICS Speed of light i vacuum, deoted by c, is equal to m/s approximately. Light is electromagetic wave (proposed by Maxwell). It cosists of varyig electric field ad magetic field. Light carries eergy ad mometum. The formula v fλ is applicable to light. (v) Whe light gets reflected i same medium, it suffers o chage i frequecy, speed ad wavelegth. (vi) Frequecy of light remais uchaged whe it gets reflected or refracted. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page

2 3. REFLECTION OF LIGHT Whe light rays strike the boudary of two media such as air ad glass, a part of light is tured back ito the same medium. This is called Reflectio of Light. (a a ) Regular Reflectio: Whe the reflectio takes place from a perfect plae surface it is called Regular Reflectio. I this case the reflected light has large itesity i oe directio ad egligibly small itesity i other directios. (b b ) Diffused Reflect io Whe the surface is rough, we do ot get a regular behavior of light. Although at each poit light ray gets reflected irrespective of the overall ature of surface, differece is observed because eve i a arrow beam of light there are may rays which are reflected from differet poits of surface ad it is quite possible that these rays may move i differet directios due to irregularity of the surface. This process eables us to see a object from ay positio. Such a reflectio is called as diffused reflectio. For example reflectio from a wall, from a ews paper etc. This is why you ca ot see your face i ews paper ad i the wall. 3. Laws of Reflectio (a) The icidet ray, the reflected ray ad the ormal at the poit of icidece lie i the same plae. This plae is called the plae of icidece (or plae of reflectio). This coditio ca be expressed mathematically as R. ( Ι N ) N. ( Ι R ) Ι. ( N R ) 0 where Ι, N (b) ad R are vectors of ay magitude alog icidet ray, the ormal ad the reflected ray respectively. The agle of icidece (the agle betwee ormal ad the icidet ray) ad the agle of reflectio (the agle betwee the reflected ray ad the ormal) are equal, i.e. i r Special Cases : Normal Icidece : I case light is icidet ormally, i r 0 δ 80º Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page

3 Ex. Grazig Icidece : I case light strikes the reflectig surface tagetially, i r 90 δ 0º or 360º Show that for a light ray icidet at a agle i o gettig reflected the agle of deviatio is δ π i or π + i. From figure (b) it is clear that light ray beds either by δ aticlockwise or by δ ( π δ ) clockwise. From figure (a) δ π i. δ π + i. 3. Object ad Image Object is defied as poit of itersectio of icidet rays. Image is defied as poit of itersectio of reflected rays (i case of reflectio) or refracted rays (i case of refractio). Let us call the side i which icidet rays are preset as icidet side ad the side i which reflected ( refracted) rays are preset, as reflected ( refracted) side. A object is called real if it lies o icidet side otherwise it is called virtual. A image is called real if it lies o reflected or refracted side otherwise it is called virtual. 4. PLANE MIRROR Plae mirror is formed by polishig oe surface of a plae thi glass plate.it is also said to be silvered o oe side. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 3

4 Ex. Sol: A beam of parallel rays of light, icidet o a plae mirror will get reflected as a beam of parallel reflected rays. For a fixed icidet light ray, if the mirror be rotated through a agle θ (about a axis which lies i the plae of mirror ad perpedicular to the plae of icidece), show that the reflected ray turs through a agle θ i same sese. See figure M, N ad R idicate the iitial positio of mirror, iitial ormal ad iitial directio of reflected light ray respectively. M, N ad R idicate the fial positio of mirror, fial ormal ad fial directio of reflected light ray respectively. From figure it is clear that ABC φ + δ (φ + θ ) or δ θ. 4. Poit object Ex.3 Characteristics of image due to Reflectio by a Plae Mirror : (i) Distace of object from mirror Distace of image from the mirror. All the icidet rays from a poit object will meet at a sigle poit after reflectio from a plae mirror which is called image. (ii) The lie joiig a poit object ad its image is ormal to the reflectig surface. (iii) The size of the image is the same as that of the object. (iv) For a real object the image is virtual ad for a virtual object the image is real Figure shows a poit object A ad a plae mirror MN. Fid the positio of image of object A, i mirror MN, by drawig ray diagram. Idicate the regio i which observer s eye must be preset i order to view the image. (This regio is called field of view). See figure, cosider ay two rays emaatig from the object. N ad N are ormals; i r ad i r The meetig poit of reflected rays R ad R is image A. Though oly two rays are cosidered it must be uderstood that all rays from A reflect from mirror such that their meetig poit is A. To obtai the regio i Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 4

5 Ex.4 which reflected rays are preset, joi A with the eds of mirror ad exted. The followig figure shows this regio as shaded. I figure there are o reflected rays beyod the rays ad, therefore the observers P ad Q caot see the image because they do ot receive ay reflected ray.] Fid the regio o Y axis i which reflected rays are preset. Object is at A (, 0) ad MN is a plae mirror, as show. The image of poit A, i the mirror is at A (6, 0). Joi A M ad exted to cut Y axis at M ( Ray origiatig from A which strikes the mirror at M gets reflected as the ray MM which appears to come from A ). Joi A N ad exted to cut Y axis at N ( Ray origiatig from A which strikes the mirror at N gets reflected as the ray NN which appears to come from A ). From Geometry. M (0, 6) N (0, 9). M N is the regio o Y axis i which reflected rays are preset. Q. See the followig figure. Which of the object(s) show i figure will ot form its image i the mirror. 4. Exteded object : A exteded object like AB show i figure is a combiatio of ifiite umber of poit objects from A to B. Image of every poit object will be formed idividually ad thus ifiite images will be formed. A will be image of A, C will be image of C, B will be image of B etc. All poit images together form exteded image. Thus exteded image is formed of a exteded object. Properties of image of a exteded object, formed by a plae mirror : () Size of exteded object size of exteded image. () The image is upright, if the exteded object is placed parallel to the mirror. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 5 (3) The image is iverted if the exteded object lies perpedicular to the plae mirror.

6 Ex.5 (4) If a exteded horizotal object is placed ifrot of a mirror iclied 45º with the horizotal, the image formed will be vertical. See figure. Show that the miimum size of a plae mirror, required to see the full image of a observer is half the size of that observer. See the followig figure. It is self explaatory if you cosider legths x ad y as show i figure. Aliter : E M, M ad E H F are similar M M H F z z or M M H F / HF / Note that the height of the mirror is half the height of eye as show i figure. Q. Figure shows a object AB ad a plae mirror MN placed parallel to object. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 6

7 Ex.6 Ex.7 Idicate the mirror legth required to see the image of object if observer s eye is at E Relatio betwee velocity of object ad image : From mirror property : x im - x om, y im y om ad z im z om Here x im meas x coordiate of image with respect to mirror. Similarly others have meaig. Differetiatig w.r.t time, we get v (im)x -v (om)x ; v (im)y v (om)y ; v (im)z v (om)z, v ig v mg (v og v mg ) for x axis but v ig v mg (v og v mg ) or v ig v og for y axis ad z axis. here: v ig velocity of image with respect to groud. A object moves with 5 m/s towards right while the mirror moves with m/s towards the left as show. Fid the velocity of image. Take as + directio. v i - v m v m - v 0 v i - (-) (-) - 5 v i - 7m/s. 7 m/s ad directio towards left. There is a poit object ad a plae mirror. If the mirror is moved by 0 cm away from the object fid the distace which the image will move. We kow that x im - x om or x i x m x m x o or x i x m x m x o. I this Q. x o 0 ; x m 0 cm. Therefore x i x m x o 0 cm. or Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 7

8 Q.3 A object is kept fixed i frot of a plae mirror which is moved by 0 m/s away from the object, fid the velocity of the image. Ex.8 I the situatio show i figure,fid the velocity of image. Sol: Alog x directio, applyig v i v m (v 0 v m ) v i ( 5cos 30º) (0 cos 60º ( 5 cos 30 0 )) Ex.9 v i - 5 (+ 3 ) m/s Alog y directio v 0 v i v i 0 si 60 5 m/s Velocity of the image - 5 ( + 3 ) î + 5 ĵ m/s. 4.4 Images formed by two plae mirrors : If rays after gettig reflected from oe mirror strike secod mirror, the image formed by first mirror will fuctio as a object for secod mirror, ad this process will cotiue for every successive reflectio. Figure shows a poit object placed betwee two parallel mirrors. Its distace from M is cm ad that from M is 8 cm. Fid the distace of images from the two mirrors cosiderig reflectio o mirror M first. To uderstad how images are formed see the followig figure ad table. You will require to kow what symbols like Ι stads for. See the followig diagram. Icidet rays Reflected by Reflected rays Object Image Object distace Image distace Rays M Rays O Ι AO cm AΙ cm Rays M Rays 3 Ι Ι BΙ cm BΙ cm Rays 3 M Rays 4 Ι Ι AΙ cm AΙ cm Rays 4 M Rays 5 Ι Ι BΙ 3cm BΙ 3cm Ad so o.. Similarly images will be formed by the rays strikig mirror M first. Total umber of images. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 8

9 Ex.0 Cosider two perpedicular mirrors. M ad M ad a poit object O. Takig origi at the poit of itersectio of the mirrors ad the coordiate of object as (x, y), fid the positio ad umber of images. Rays a ad b strike mirror M oly ad these rays will form image Ι at (x, y), such that O ad I are equidistat from mirror M. These rays doot form further image because they do ot strike ay mirror agai. Similarly rays d ad e strike mirror M oly ad these rays will form image Ι at ( x, y), such that O ad I are equidistat from mirror M. Now cosider those rays which strike mirror M first ad the the mirror M. For icidet rays, object is O, ad reflected rays 3, 4 form image Ι. Now rays 3, 4 icidet o M (object is Ι ) which reflect as rays 5, 6 ad form image Ι. Rays 5, 6 do ot strike ay mirror, so image formatio stops. Ι ad Ι, are equidistat from M. To summarize see the followig figure Now rays 3, 4 icidet o M (object is Ι ) which reflect as rays 5, 6 ad form image Ι. Rays 5, 6 do ot strike ay mirror, so image formatio stops. For rays reflectig first from M ad the from M, first image Ι (at (x, y)) will be formed ad this will fuctio as object for mirror M ad the its image Ι (at ( x, y)) will be formed. I ad I coicide. Three images are formed Q.4 Figure shows two iclied plae mirrors M ad M ad a object O. Its images formed i mirrors M ad M idividually are Ι ad Ι respectively. Show that Ι ad Ι ad O lie o the circumferece of a circle with cetre at O. [This result ca be exteded to show that all the images will also lie o the same circle. Note that this result is idepedet of the agle of icliatio of mirrors.] 4. 5 Locatig all the Images formed by two Plae Mirrors Cosider two plae mirrors M ad M iclied at a agle θ α+β as show i figure Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 9

10 Ex. Poit P is a object kept such that it makes agle α with mirror M ad agle β with mirror M. Image of object P formed by M,deoted by Ι,will be iclied by agle α o the other side of mirror M. This agle is writte i bracket i the figure besides Ι. Similarly image of object P formed by M,deoted by Ι,will be iclied by agle β o the other side of mirror M. This agle is writte i bracket i the figure besides Ι. Now Ι will act as a object for M which is at a agle (α+β) from M. Its image will be formed at a agle (α+β) o the opposite side of M. This image will be deoted as Ι, ad so o. Thik whe this will process stop. Hit: The virtual image formed by a plae mirror must ot be i frot of the mirror or its extesio. Number of images formed by two iclied mirrors (i) If (ii) If bisector. (iii) If bisector. (iv) If 360 º 360 º eve umber; umber of image θ θ 360 º 360 º odd umber; umber of image, if the object is placed o the agle θ θ 360 º 360 º odd umber; umber of image, if the object is ot placed o the agle θ θ 360º iteger, the cout the umber of images as explaied above. θ Two mirrors are iclied by a agle A object is placed makig 0 0 with the mirror M. Fid the positios of first two images formed by each mirror. Fid the total umber of images usig (i) direct formula ad (ii) coutig the images. Figure is self explaatory. Number of images (i) Usig direct formula : (ii) By coutig. See the followig table 360 º (eve umber) 30º umber of images Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 0

11 5. SPHERICAL MIRRORS Ex. Ex.3 cos i Spherical Mirror is formed by polishig oe surface of a part of sphere. Depedig upo which part is shiig the spherical mirror is classified as (a) Cocave mirror, if the side towards ceter of curvature is shiig ad (b) Covex mirror if the side away from the ceter of curvature is shiig. A poit o the surface o the mirror from where the positio of the object ca be specified easily is called pole. The pole is geerally take at the mid poit of reflectig surface. The cetre of the sphere of which the mirror is a part, is called cetre of curvature.the radius of the sphere of which the mirror is a part is called Radius of curvature.the straight lie coectig pole P ad cetre of curvature C is Pricipal Axis. Fid the agle of icidece of ray for which it passes through the pole, give that MΙ CP. MIC CΙP θ MΙ CP MΙθ ΙCP θ CΙ CP CΙP CPΙ θ I CΙP all agle are equal 3θ 80º θ 60º Fid the distace CQ if icidet light ray parallel to pricipal axis is icidet at a agle i. Also fid the distace CQ if i 0. R CQ R CQ cosi As i icreases cos i decreases. Hece CQ icreases Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page

12 If i is a small agle cos i CQ R/ So, paraxial rays meet at a distace equal to R / from ceter of curvature, which is called focus. Pricipal focus (F) is the poit of itersectio of all the reflected rays for which the icidet rays strike the mirror (with small aperture) parallel to the pricipal axis. I cocave mirror it is real ad i the covex mirror it is virtual. The distace from pole to focus is called focal legth. Aperture (related to the size of mirror) is the diameter of the mirror. Covex mirror Cocave mirror 5. Ray tracig : Followig facts are useful i ray tracig. (i). If the icidet ray is parallel to the pricipal axis, the reflected ray passes through the focus. (ii). If the icidet ray passes through the focus, the the reflected ray is parallel to the pricipal axis. (iii). (iv). (i). Icidet ray passig through cetre of curvature will be reflected back through the cetre of curvature (because it is a ormally icidet ray). It is easy to make the ray tracig of a ray icidet at the pole as show i below. 5. Sig Covetio We are usig co ordiate sig covetio. Take origi at pole (i case of mirror )or at optical cetre (i case of les) Take X axis alog the Pricipal Axis,takig positive directio alog the icidet light. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page

13 (ii). Ex.4 Ex.5 u, v, R ad f idicate the x coordiate of object, image, cetre of curvature ad focus respectively. y-coordiates are take positive above Pricipal Axis ad egative below Pricipal Axis h ad h deote the y coordiate of object ad image respectively. Note : This sig covetio is used for reflectio from mirror, reflectio through flat or curved surfaces or les. 5.3 Formulae for Reflectio from spherical mirrors : 5.3. Mirror formula : v + u R f X-coordiate of cetre of Curvature ad focus of Cocave mirror are egative ad those for Covex mirror are positive. I case of mirrors sice light rays reflect back i X-directio, therefore -ve sig of v idicates real image ad +ve sig of v idicates virtual image. Figure shows a spherical cocave mirror with its pole at (0, 0) ad pricipal axis alog x axis. There is a poit object at ( 40 cm, cm), fid the positio of image. Accordig to sig covetio, u 40 cm h + cm f 5 cm. v v + u h f + ; v 40 5 v u cm cm. ; h 40 h h 40 The positio of image is cm, cm 7 7 v u Covergig rays are icidet o a covex spherical mirror so that their extesios itersect 30 cm behid the mirror o the optical axis. The reflected rays form a divergig beam so that their extesios itersect the optical axis. m from the mirror. Determie the focal legth of the mirror. I this case u + 30 v + 0 Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 3

14 + f v u 0 f 4 cm + 30 Ex.6 Fid the positio of fial image after three successive reflectios takig first reflectio o m. Ι reflectio : Focus of mirror 0 cm u 5 cm Applyig mirror formula : v + u f v 30 cm. For ΙΙ reflectio o plae mirror : u 0 cm v 0 cm For ΙΙΙ reflectio o curved mirror agai : u 50 cm f 0 cm Applyig mirror formula : v + u f v.5 cm. Q.5 Fid the positio of fial image after three successive reflectios takig first reflectio o m. Q.6 Fid the positio of fial image after three successive reflectios takig first reflectio o m Lateral magificatio (or trasverse magificatio) deoted by m is defied as m h ad is related as h m v.from the defiitio of m positive sig of m idicates erect image ad egative sig idicates u iverted image. From the defiitio of m positive sig of m idicates erect image ad egative sig idicates iverted image I case of successive reflectio from mirrors, the overall lateral magificatio is give by m m m 3..., where m, m etc. are lateral magificatios produced by idividual mirrors. Note : Usig (5.3.) ad (5.3.) the followig coclusios ca be made (check yourself). Nature of Object Nature of Image Iverted or erect Real Real Iverted Real Virtual Erect Virtual Real Erect Virtual Virtual Iverted Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 4

15 f f v From (5.3.) ad (5.3.); we get m...(just a time savig formula) f u f Ex.7 A exteded object is placed perpedicular to the pricipal axis of a cocave mirror of radius of curvature 0 cm at a distace of 5 cm from pole. Fid the lateral magificatio produced. u 5 cm f 0 cm Ex.8 Usig m + we get, v 30 cm v u f v. u f Aliter : m f u 0 0 ( 5) A perso looks ito a spherical mirror. The size of image of his face is twice the actual size of his face. If the face is at a distace 0 cm the fid the ature of radius of curvature of the mirror. Perso will see his face oly whe the image is virtual. Virtual image of real object is erect. Hece m Applyig v u Aliter : m v 40 cm + ; f 40 cm or R 80 cm. v u f f f u f f ( 0) or R 80 cm f 40 cm Ex.9 A image of a cadle o a scree is foud to be double its size. Whe the cadle is shifted by a distace 5 cm the the image become triple its size. Fid the ature ad ROC of the mirror. Sice the images formed o scree it is real. Real object ad real image implies cocave mirror. Applyig m After shiftig 3 f f u f f (u + 5) or f f (u)...()...() [Why u + 5?, why ot u 5 : I a cocave mirror are size of real image will icrease, oly whe the real object is brought closer to the mirror. I doig so, its x coordiate will icrease] From () & () we get, f 30 cm or R 60 cm Q.7 A coi is placed 0 cm i frot of a cocave mirror. The mirror produces a real image that has diameter 4 times that of the coi. What is the image distace. Q.8 A small statue has a height of cm ad is placed i frot of a spherical mirror. The image of the statue is iverted ad is 0.5cm tall ad located 0 cm i frot of the mirror. Fid the focal legth ad ature of the mirror. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 5

16 5.3.5 O differetiatig the mirror formula we get dv v du. u Mathematically 'du' implies small chage i positio of object ad 'dv' implies correspodig small chage i positio of image. If a small object lies alog pricipal axis, du may idicate the size of object ad dv the size of its image alog Pricipal axis (Note that the focus should ot lie i betwee the iitial ad fial poits of object). I this case dv du is called logitudial magificatio. Negative sig idicates iversio of image irrespective of ature of image ad ature of mirror. Ex.0 A poit object is placed 60 cm from pole of a cocave mirror of focal legth 0 cm o the pricipal axis. Fid (a) the positio of image (b) If object is shifted mm towards the mirror alog pricipal axis fid the shift i image. Explai the result. (a) u 60 cm f 0cm (b) (A) (B) fu v u f v + u f 0( 60) 60 ( 0) cm. v Differetiatig, we get dv du [ mm ] mm u 60 5 [ du mm; sig of du is + because it is shifted i +ve directio defied by sig covetio.] ve sig of dv idicates that the image will shift towards egative directio. The sig of v is egative. Which implies the image is formed o egative side of pole. (A) ad (B) together imply that the image will shift away from pole. Note that differetials dv ad du deote small chages oly Velocity of image (a) Object movig perpedicular to pricipal axis : From the relatio i 5.3., h we have h v u or h v. h u If a poit object moves perpedicular to the pricipal axis, x coordiate of both the object & the image become costat. O differetiatig the above relatio w.r.t. time, we get, dh Here, dt dh dt dh u dt v dh deotes velocity of object perpedicular to the pricipal axis ad dt perpedicular to the pricipal axis. deotes velocity of image (b) Object movig alog pricipal axis : O differetiatig the mirror formula with respect to time we get Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 6

17 dv v du dv dt, where u dt dt is the velocity of image alog Pricipal axis ad du dt is the velocity of object alog Pricipal axis. Negative sig implies that the image, i case of mirror, always moves i the directio opposite to that of object. This discussio is for velocity with respect to mirror ad alog the x axis. (c) Object movig at a agle with the pricipal axis : Resolve the velocity of object alog ad perpedicular to the pricipal axis ad fid the velocities of image i these directios separately ad the fid the reusltat Newto's Formula: XY f X ad Y are the distaces ( alog the pricipal axis ) of the object ad image respectively from the pricipal focus. This formula ca be used whe the distaces are metioed or asked from the focus Optical power of a mirror (i Diopters) f f focal legth with sig ad i meters If object lyig alog the pricipal axis is ot of very small size, the logitudial magificatio v u v u 6. REFRACTION OF LIGHT (it will always be iverted) Deviatio or bedig of light rays from their origial path while passig from oe medium to aother is called refractio. It is due to chage i speed of light as light passes from oe medium to aother medium. If the light is icidet ormally the it goes to the secod medium without bedig, but still it is called refractio. Refractive idex of a medium is defied as the factor by which speed of light reduces as compared to the c speed of light i vacuum speed of light i vacuum. µ v speed of light i medium. More (less) refractive idex implies less (more) speed of light i that medium, which therefore is called deser (rarer) medium. 6. Laws of Refractio (a) The icidet ray, the ormal to ay refractig surface at the poit of icidece ad the refracted ray all lie i the same plae called the plae of icidece or plae of refractio. (b) Si i Sir Law. Also, Costat for ay pair of media ad for light of a give wave legth. This is kow as Sell's Si i Si r v v For applyig i problems remember sii sir λ λ Refractive Idex of the secod medium with respect to the first medium. C speed of light i air (or vacuum) 3 x 0 8 m/s. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 7

18 Special cases : (i) Normal icidece : i 0 from sell s law : r 0 (ii) (iii) Whe light moves from deser to rarer medium it beds away from ormal. Whe light moves from rarer to deser medium it beds towards the ormal. Note: (i) Higher the value of R.Ι., deser (optically) is the medium. (ii) Frequecy of light does ot chage durig refractio. Ex. (iii) Refractive idex of the medium relative to vacuum µ r r vacuum ; air ~ > ; water (average value) 4/3 ; glass (average value) 3/ 6. Deviatio of a Ray Due to Refractio Deviatio (δ) of ray icidet at i ad refracted at r is give by δ i r. A light ray is icidet o a glass sphere at a agle of icidece 60 0 as show. Fid the agles r, r,e ad the total deviatio after two refractios. Applyig Sell s law si sir r 30 0 Ex. e 60 0 From symmetry r r Agai applyig sell s law at secod surface si e 3 sir Deviatio at first surface i r Deviatio at secod surface e r Therefore total deviatio Fid the agle θ a made by the light ray whe it gets refracted from water to air, as show i figure. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 8

19 Sell s Law Ex µ W si θ W µ a si θ a si θ 3 5 a si θ a θ a si 5 Fid the speed of light i medium a if speed of light i medium b is 3 c where c speed of light i vacuum ad light refracts from medium a to medium b makig 45º ad 60º respectively with the ormal. Sell s Law µ a si θ a µ b si θ b c si θa v a v a c si θb. v b c c si 45º si 60º. c / 3 v a 3 c 3 Q.9 A light ray deviates by 30 0 (which is oe third of the agle of icidece) whe it gets refracted from vacuum to a medium. Fid the refractive idex of the medium. Q.0 A coi lies o the bottom of a lake m deep at a horizotal distace x from the spotlight (a source of thi parallel beam of light) situated m above the surface of a liquid of refractive idex µ ad height m. Fid x. 6.3 Priciple of Reversibility of Light Rays (a) A ray travellig alog the path of the reflected ray is reflected alog the path of the icidet ray. (b) A refracted ray reversed to travel back alog its path will get refracted alog the path of the icidet ray. Thus the icidet ad refracted rays are mutually reversible. (c) Accordig to this priciple. 7. REFRACTION THROUGH A PARALLEL SLAB Note: Whe light passes through a parallel slab, havig same medium o both sides, the (a) Emerget ray is parallel to the icidet ray. Emerget ray will ot be parallel to the icidet ray if the medium o both the sides of slab are differet. (b) Light is shifted laterally, give by (studet should be able to derive it) tsi(i r) d cosr t thickess of slab Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 9

20 Ex.4 Fid the lateral shift of light ray while is passes through a parallel glass slab of thickess 0 cm placed i air. The agle of icidece i air is 60º ad the agle of refractio i glass is 45º. t si (i r) d cosr 0si (60 45 ) cos45 0si 5 0 cos45 5 si 5º. Q. A ray of light falls at a agle of 30º oto a plae-parallel glass plate ad leaves it parallel to the iitial ray. The refractive idex of the glass is.5. What is the thickess d of the plate if the distace betwee the rays is 3.8 cm? [Give : si 9.5º ; cos 9.5º 0.94 ; si 0.5º 0.8] 3 Q. A light passes through may parallel slabs oe by oe as show i figure. Prove that sii sii 3 sii 3 4 sii 4...[Remember this]. Also prove that if 4 the light rays i medium ad i medium 4 are parallel. 7. Apparet Depth ad shift of Submerged Object At ear ormal icidece (small agle of icidece i) apparet depth (d ) is give by: d d relative where relative i(r. Ι.of medium of icidece) (R. Ι.of medium of refractio) r d distace of object from the iterface real depth d distace of image from the iterface apparet depth This formula ca be easily derived usig sell s law ad applyig the coditio of early ormal icidece... (try it). Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 0

21 Ex.5 Apparet shift d rel A object lies 00 cm iside water.it is viewed from air early ormally. Fid the apparet depth of the object. Sol: d d relative cm 4 / 3 Q.3 A object lies 90 cm i air above water surface.it is viewed from water early ormally. Fid the apparet height of the object. Ex.6 A cocave mirror is placed iside water with its shiig surface upwards ad pricipal axis vertical as show. Rays are icidet parallel to the pricipal axis of cocave mirror. Fid the positio of fial image. The icidet rays will pass udeviated through the water surface ad strike the mirror parallel to its pricipal axis. Therefore for the mirror, object is at. Its image A (i figure) will be formed at focus which is 0 cm from the mirror. Now for the iterface betwee water ad air, d 0 cm. d 0 d w 4 / cm. a Q.4 A cocave mirror is placed iside water with its shiig surface upwards ad pricipal axis vertical as show. Rays are icidet parallel to the pricipal axis of cocave mirror. Fid the positio of fial image. Q.5 Prove that the shift i positio of object due to parallel slab is give by shift d where rel rel. ' Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page

22 Ex.7 See the figure (i) (ii) (iii) (iv) (i) d B Ex.8 (ii) d F (iii) (iv) See the figure. Fid apparet height of the bird Fid apparet depth of fish At what distace will the bird appear to the fish. At what distace will the fish appear to the bird / cm 4 / 3 48 cm For fish : d B cm d B cm For bird : d F cm. d F cm. Fid the distace of fial image formed by mirror Shift 3 3/ 3 3/ For mirror object is at a distace 3 3 / 0 cm Object is at the cetre of curvature of mirror. Hece the light rays will retrace ad image will formed o the object itself. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page

23 7. Refractio through a Composite Slab (or Refractio through a umber of parallel media, as see from a medium of R. I. 0 ) Ex.9 Apparet depth (distace of fial image from fial surface) Apparet shift t t rel + t rel + t 3 3rel t rel + t rel t rel rel Where ' t ' represets thickess ad ' ' represets the R.I. of the respective media, relative to the medium of observer. (i.e. rel / 0, rel / 0 etc.) See figure. Fid the apparet depth of object see below surface AB. d 0 D app + µ cm..8 Q.6 Fid the apparet depth of object O below surface AB, see by a observer i medium of refractive idex µ Q.7 I above questio what is the depth of object correspodig to icidet rays strikig o surface CD i medium µ. Q.8 I above questio if observer is i medium µ 3, what is the apparet depth of object see below surface CD. 8. CRITICAL ANGLE AND TOTAL INTERNAL REFLECTION ( T. T. I. R.) Critical agle is the agle made i deser medium for which the agle of refractio i rarer medium is 90º. Whe agle i deser medium is more the critical agle the light ray reflects back i deser medium followig the laws of reflectio ad the iterface behaves like a perfectly reflectig mirror. I the figure O Object NN Normal to the iterface II Iterface C Critical agle; AB reflected ray due to T. I. R. Whe i C the r 90 o Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 3

24 C si r d Ex.30 Ex.3 8. Coditios of T. I. R. (a) light is icidet o the iterface from deser medium. (b) Agle of icidece should be greater tha the critical agle (i > c). Figure shows a lumious object placed i deser medium at a distace h from a iterface separatig two media of refractive idices µ r ad µ d. Subscript r &d stad for rarer ad deser medium respectively. I the figure ray strikes the surface at a agle less tha critical agle C ad gets refracted i rarer medium. Ray strikes the surface at critical agle ad grazes the iterface. Ray 3 strikes the surface makig a agle more tha critical agle ad gets iterally reflected. The locus of poits where ray strikes at critical agle is a circle, called circle of illumiace. All light rays strikig iside the circle of illumiace get refracted i rarer medium. If a observer is i rarer medium, he/she will see light comig out oly from withi the circle of illumiace. If a circular opaque plate covers the circle of illumiace, o light will get refracted i rarer medium ad the the object ca ot be see from the rarer medium. Radius of C.O.I ca be easily foud. Fid the max. agle that ca be made i glass medium (µ.5) if a light ray is refracted from glass to vacuum..5 si C si 90º, where C critical agle. si C /3 C si /3 Fid the agle of refractio i a medium (m ) if light is icidet i vacuum, makig agle equal to twice the critical agle. Sice the icidet light is i rarer medium. Total Iteral Reflectio ca ot take place. C si µ 30º i C 60º Applyig Sell s Law. si 60º si r si r 3 4 r si 3 4. Q.9 Fid the radius of C.O.I., If a lumious object is placed at a distace h from the iterface i deser medium. Q.0 A ship is sailig i river. A observer is situated at a depth h i water (µ w ). If x >> h, fid the agle made from vertical, of the lie of sight of ship. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 4

25 Ex.3 Ex.33 Sol: What should be the value of agle θ so that light eterig ormally through the surface AC of a prism (3/) does ot cross the secod refractig surface AB. Light ray will pass the surface AC without bedig sice it is icidet ormally. Suppose it strikes the surface AB at a agle of icidece i. i 90-θ For the required coditio: 90 θ >C or si (90 θ) > sic or cosθ > sic 3/ 3 or θ < cos -. 3 What should be the value of refractive idex of a glass rod placed i air, so that the light eterig through the flat surface of the rod does ot cross the curved surface of the rod. It is required that all possible r should be more tha critical agle. This will be automatically fulfilled if miimum r is more tha critical agle...(a) Agle r is miimum whe r is maximum i.e. C( why?).therefore the miimum value of r is 90-C. From coditio (A) : 90 C > C or C < 45 si C < si 45 ; 9. CHARACTERISTICS OF A PRISM (a) (b) (c) (d) (e) (f) < or >. A homogeeous solid trasparet ad refractig medium bouded by two plae surfaces iclied at a agle is called a prism. 3-D view Refractio through a prism: View from oe side PQ ad PR are refractig surfaces. QPR A is called refractig agle or the agle of prism (also called Apex agle). δ agle of deviatio For refractio of a moochromatic ( sigle wave legth) ray of light through a prism; δ (i + e) (r + r ) ad r + r A δ i + e A. Variatio of δ versus i (show i diagram). For oe δ (except δ mi) there are two values of agle of icidece. If i ad e are iterchaged the we get the same value of δ because of reversibility priciple of light Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 5

26 Ex.34 (j) (i) (g) There is oe ad oly oe agle of icidece for which the agle of deviatio is miimum. (h) Whe δ δ mi, the agle of miimum deviatio, the i e ad r r, the ray passes symmetrically w.r.t. the refractig surfaces. We ca show by simple calculatio that δ mi i mi A where i mi agle of icidece for miimum deviatio, ad r A/. (i) si rel si A + δm [ A ], where rel Also δ mi ( ) A (for small values of A) prism surroudigs For a thi prism ( A 0 o ) ad for small value of i, all values of δ ( ) A Show that if A > A max ( C), the Total iteral reflectio occurs at secod refractig surface PR for ay value of ' i '. For T.I.R. at secod surface r > C (A r) > C or A > (C + r) The above relatio will be fullfilled if or A > C + r max or A > C + C or A > C O the basis of above example ad similar reasoig, it ca be show that (you should try the followig cases (ii) ad (iii) yourself.) If A > C, all rays are reflected back from the secod surface. (ii) If A C, o rays are reflected back from the secod surface i.e. all rays are refracted from secod surface. (iii) If C A > C, some rays are reflected back from the secod surface ad some rays are refracted from secod surface, depedig o the agle of icidece.. (k) δ is maximum for two values of i i mi (correspodig to e 90º) Ex.35 Ex.36 ad i 90º (correspodig to e mi ). For i mi : s si i mi p si(a C) If i < i mi the T.I.R. takes place at secod refractig surface PR. Refractig agle of a prism A 60º ad its refractive idex is, 3/, what is the agle of icidece i to get miimum deviatio. Also fid the miimum deviatio. Assume the surroudig medium to be air ( ). For miimum deviatio, r r A 30º. applyig sell s law at I surface 3 3 si i si 30º i si 4 3 π δ mi si 4 3 See the figure Fid the deviatio caused by a prism havig refractig agle 4º ad refractive idex 3. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 6

27 δ ( 3 ) 40 0 Ex.37 For a prism, A 60º, Sol 7. Fid the miimum possible agle of icidece, so that the light ray is refracted 3 from the secod surface. Also fid δ max. I miimum icidece case the agles will be as show i figure Applyig sell s law : si i mi 7 si ( A C) 3 7 (si A cos C cos A si C) i mi 30º δ max i mi A Q. Fid r, r, e, δ for the case show i figure. 3 3 si 60 cos Q. For the case show i figure prove the relatios r r A ad δ (i e) + A (do ot try to remember these relatios because the prism is ormally ot used i this way). Q.3 From the graph of agle of deviatio δ versus agle of icidece i, fid the prism agle 0. DISPERSION OF LIGHT The agular splittig of a ray of white light ito a umber of compoets ad spreadig i differet directios is called Dispersio of Light. [It is for whole Electro Magetic Wave i totality]. This pheomeo is Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 7

28 Ex.38 because waves of differet wavelegth move with same speed i vacuum but with differet speeds i a medium. Therefore, the refractive idex of a medium depeds slightly o wavelegth also. This variatio of refractive idex with wavelegth is give by Cauchy s formula. b Cauchy's formula (λ) a + where a ad b are positive costats of a medium. λ Note : Such pheomeo is ot exhibited by soud waves. Agle betwee the rays of the extreme colours i the refracted (dispersed) light is called agle of dispersio. θ δ v δ r (Fig. (a)) Fig (a) ad (c) represets dispersio, whereas i fig. (b) there is o dispersio. For prism of small A ad with small i : θ δ v δ r ( v r )A The refractive idices of flit glass for red ad violet light are.63 ad.63 respectively. Fid the agular dispersio produced by a thi prism of flit glass havig refractig agle 5 0. Deviatio of the red light is δ r (µ r )A ad deviatio of the violet light is δ v (µ v )A. The disperatio δ v δ r (µ v µ r )A (.63.63) Deviatio of beam (also called mea deviatio) δ δ y ( y )A v, r ad y are R. Ι. of material for violet, red ad yellow colours respectively. Note : Numerical data reveals that if the average value of µ is small µ v µ r is also small ad if theaverage value of µ is large µ v µ r is also large. Thus, larger the mea deviatio, larger will be the agular dispersio. Ex.39 Dispersive power (ω) of the medium of the material of prism is give by: v r ω y Note : ω is the property of a medium. For small agled prism ( A 0 o v r δv δ ) with light icidet at small agle i : y δy agular dispersio deviatio of mea ray (yellow) [ y v + r if y is ot give i the problem ] Note: refractivity of the medium for the correspodig colour. Refractive idex of glass for red ad violet colours are.50 ad.60 respectively. Fid (A) the ref. idex for yellow colour, approximately (B) Dispersive power of the medium. r θ δ y Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 8

29 µ v + µ (A) µ r ~ R Ex.40 Q (B) ω µ v µ µ 0. Dispersio without deviatio (Direct Visio Combiatio) The coditio for direct visio combiatio is : [ y ] A [ y ] v + r A A r R v + r Two or more prisms ca be combied i various ways to get differet combiatio of agular dispersio ad deviatio.6 0. Deviatio without dispersio (Achromatic Combiatio ) Coditio for achromatic combiatio is: ( v r ) A ( v r ) A If two prisms are combied, as show i figure, fid the total agular dispersio ad agle of deviatio suffered by a white ray of light icidet o the combiatio. Both prisms will tur the light rays towards their bases ad hece i same directio. Therefore turigs caused by both prisms are additive. Total agular dispersio θ + θ (µ V µ R ) A + (µ V µ R ) A (.5.4) 4º + (.7.5)º 0.8 Total derivatio δ + δ µ V + µ R µ ' A + V +µ ' R A º + 0.º (.45 ) 0.4º + (.6 ) 0.º º º º As. If two prisms are combied, as show i figure, fid the et agular dispersio ad agle of deviatio suffered by a white ray of light icidet o the combiatio. A Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 9

30 Ex.4 Two thi prisms are combied to form a achromatic combiatio. For I prism A 4º, µ R.35, µ Y.40, µ v.4. for II prism µ R.7, µ Y.8 ad µ R.9 fid the prism agle of II prism ad the et mea derivatio. Coditio for achromatic combiatio. θ θ (µ V µ R )A (µ V µ R )A Ex.4 (.4.35)4º A δ Net δ ~ δ (µ Y )A ~ (µ Y )A (.40 ) 4º ~ (.8 ) º. θ θ (µ V µ R )A (µ V µ R )A (.4.35)4º A δ Net δ ~ δ (µ Y )A ~ (µ Y )A (.40 ) 4º ~ (.8 ) º. A crow glass prism of agle 5 0 is to be combied with a flit prism i such a way that the mea ray passes udeviated. Fid (a) the agle of the flit glass prism eeded ad (b) the agular dispersio produced by the combiatio whe white light goes through it. Refractive idices for red, yellow ad violet light are.54,.57 ad.53 respectively for crow glass ad.63,.60 ad.63 for flit glass. The deviatio produced by the crow prism is δ (µ )A ad by the flit prism is : δ' (µ' )A'. The prisms are placed with their agles iverted with respect to each other. The deviatios are also i opposite directios. Thus, the et deviatio is : D δ δ' (µ )A (µ' )A'....() (a) If the et deviatio for the mea ray is zero, (µ )A (µ' )A'. or, A' ( µ ) A ( µ ' ) (b) The agular dispersio produced by the crow prism is : δ v δ r (µ v µ r )A ad that by the flit prism is, δ' v δ' r (µ' v µ' r )A The et agular dispersio is, (µ v µ r )A (µ' v µ' r )A (.53.54) 5 0 (.63.63) The agular dispersio has magitude (a) If the et deviatio for the mea ray is zero, (µ )A (µ' )A'. or, A' ( µ ) A ( µ ' ) (b) The agular dispersio produced by the crow prism is : δ v δ r (µ v µ r )A 0 0 Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 30

31 ad that by the flit prism is, δ' v δ' r (µ' v µ' r )A The et agular dispersio is, (µ v µ r )A (µ' v µ' r )A (.53.54) 5 0 (.63.63) The agular dispersio has magitude Q.5 The dispersive powers of crow ad flit glasses are 0.03 ad 0.05 respectively. The refractive idices for yellow light for these glasses are.57 ad.6 respectively. It is desired to form a achromatic combiatio of prisms of crow ad flit glasses which ca produce a deviatio of 0 i the yellow ray. Fid the refractig agles of the two prisms eeded.. SPECTRUM : (Oly for your kowledge ad ot of much use for JEE) Ordered patter produced by a beam emergig from a prism after refractio is called Spectrum. Types of spectrum:. Types of spectrum: (a) Lie spectrum: Due to source i atomic state. (b) Bad spectrum: Due to source i molecular state. (c) Cotiuous spectrum: Due to white hot solid.. I Emissio Spectrum: Bright colours or lies, emitted from source are observed. The spectrum emitted by a give source of light is called emissio spectrum. It is a wavelegth-wise distributio of light emitted by the source. The emissio spectra are give by icadescet solids, liquids ad gases which are either bured directly as a flame (or a spark) or burt uder low pressure i a discharge tube..3 I Absorptio Spectrum: Dark lies idicates frequecies absorbed. Whe a beam of light from a hot source is passed through a substace (at a lower temperature), a part of the light is trasmitted but rest of it is absorbed. With the help of a spectrometer, we ca kow the fractio of light obsorbed correspodig to each wavelegth. The distributio of the wavelegth absorptioof light by a substace is called a absorptio spectrum. Every substace has its ow characteristic absorptio spectrum..4 Spectrometer Cosists of a collimator (to collimate light beam), prism ad telescope. It is used to observe the spectrum ad also measure deviatio.. REFRACTION AT SPHERICAL SURFACES For paraxial rays icidet o a spherical surface separatig two media: v u R... (A) where light moves from the medium of refractive idex to the medium of refractive idex. Trasverse magificatio (m) (of dimesio perpedicular to pricipal axis) due to refractio at spherical surface is give by m v R v / u R u/ Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 3

32 Ex.43 Fid the positio, size ad ature of image, for the situatio show i figure. Draw ray diagram. For refractio ear poit A, u 30 ; R 0; ;. Applyig refractio formula v u R v 30 v 60 cm m h v h u h 4 mm. 0 ( 60) 4 ( 30) Q.6 See the situatio show i figure Ex.44 Sol: () Fid the positio of image as see by observer A. () Fid the positio of image as see by observer B. Special case: Refractio at plae Surfaces Puttig R i the formula, we get; v u R u v The same sig of v ad u implies that the object ad the image are always o the same side of the iterface separatig the two media. If we write the above formula as v u rel, it gives the relatio betwee the apparet depth ad real depth, as we have see before. Usig formula of spherical surface or otherwise, fid the apparet depth of a object placed 0 cm below the water surface, if see ear ormally from air. Put R i the formula of the Refractio at Spherical Surfaces we get, v u u 0 cm 3 4 Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 3

33 0 v 7.5 cm 4 / 3 ve sig implies that the image is formed i water. Aliter: 3. Thi Les d app d real µ rel cm. 4 / 3 4 A thi les is called covex if it is thicker at the middle ad it is called cocave if it is thicker at the eds. Oe surface of a covex les is always covex. Depedig o the other surface a covex les is categorized as (a) bicovex or covexo covex, if the other surface is also covex, (b) Plao covex if the other surface is plae ad (c) Cocavo covex if the other surface is cocave. Similarly cocave les is categorized as cocavo-cocave or bicocave, plao-cocave ad covexo-cocave. For a spherical, thi les havig the same medium o both sides: les where rel surface respectively. v u ( ) rel R R medium Les has two Focii: v u...(a), ad R ad R are x coordiates of the cetre of curvature of the st surface ad d f Les Maker's Formula...(b) If u, the v f v f If icidet rays are parallel to pricipal axis the its refracted ray will cut the pricipal axis at f. It is called d focus. I case of covergig les it is positive ad i case of divergig les it is egative. If v that meas Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 33 u f u f

34 Ex.45 If icidet rays cuts pricipal axis at f the its refracted ray will become parallel to the pricipal axis. It is called st focus. I case of covergig les it is egative ( f is positive) ad i the case of divergig les it positive ( f is egative) use of f & + f is i drawig the ray diagrams. Notice that the poit B, its image B ad the pole P of the les are colliear. It is due to parallel slab ature of the les at the middle. This ray goes straight. (Remember this) From the relatio f ( rel ) R R factors. (A) The factor is R R (a) Positive for all types of covex leses ad (b) Negative for all types of cocave leses. (B) (C) (D) it ca be see that the secod focal legth depeds o two The factor ( rel ) is (a) Positive whe surroudig medium is rarer tha the medium of les. (b) Negative whe surroudig medium is deser tha the medium of les. So a les is covergig if f is positive which happes whe both the factors (A) ad (B) are of same sig. Ad a les is divergig if f is egative which happes whe the factors (A) ad (B) are of opposite sigs. Fid the behavior of a cocave les placed i a rarer medium. Factor (A) is egative, because the les is cocave. Factor (B) is positive, because the les is placed i a rarer medium. Therefore the focal legth of the les, which depeds o the product of these factors, is egative ad hece the les will behave as divergig les. Ex.46 Show that the factor (ad therefore focal legth) does ot deped o which surface of the les R R light strike first. Sol: CASE : Cosider a covex les of radii of curvature p ad q as show. Suppose light is icidet from left side ad strikes the surface with radius of curvature p, first. The R +p ; R -q ad R R F HG I F HG + p qkj p q CASE : Suppose light is icidet from right side ad strikes the surface with radius of curvature q, first. I KJ Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 34

35 The R +q ; R -p ad R R F HG I F HG I KJ + q pkj p q Though we have show the result for bicovex les, it is true for every les. Q.7 Fid the focal legth of a double-covex les with R 5 cm ad R 5 cm. The refractive idex of the les material.5. Q.8 Fid the focal legth of a plao-covex les with R 5 cm ad R. The refractive idex of the les material.5. Q.9 Fid the focal legth of a cocavo-covex les (positive meiscus) with R 5 cm ad R 5 cm. The refractive idex of the les material.5. Ex.47 Ex.48 Fid the focal legth of the les show i the figure. f + 0 cm. (rel ) f R R (3/ ) f 0 ( 0) f 0 Fid the focal legth of the les show i figure (rel ) 3 f R R 0 0 f 0 cm Ex: 49 Poit object is placed o the pricipal axis of a thi les with parallel curved boudaries i.e., havig same radii of curvature. Discuss about the positio of the image formed. Sol: Ex.50 Sol: f ( rel ) 0 [ R R R R ] v u 0 or v u i.e. rays pass without appreciable bedig. Focal legth of a thi les i air, is 0 cm. Now medium o oe side of the les is replaced by a medium of refractive idex µ. The radius of curvature of surface of les, i cotact with the medium, is 0 cm. Fid the ew focal legth. Let radius of Ι surface be R ad refractive idex of les be µ. Let parallel rays be icidet o the les. Applyig refractio formula at first surface µ V µ R...() Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 35

36 Ex.5 At ΙΙ surface Addig () ad () µ µ V V 0 µ µ µ V + V V R + (µ ) R 0 µ 0 µ 0 v 40 cm f 40 cm µ 0...() f (i air) Figure show a poit object ad a covergig les. Fid the fial image formed. v u f v 5 0 v v + 30 cm Q.30 Figure shows a poit object ad a divergig les. Fid the fial image formed. Ex.5 Ex.53 See the figure Fid the positio of fial image formed. For covergig les u 5 cm, f 0 cm fu v 30 cm f + u For divergig les u 5 cm f 0 cm fu v 0 cm f + u Figure shows two covergig leses. Icidet rays are parallel to pricipal axis. What should be the value of d so that fial rays are also parallel. Fial rays should be parallel. For this the ΙΙ focus of L must coicide with Ι focus of L. d cm Here the diameter of ray beam becomes wider. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 36

37 Ex.54 See the figure Ex.55 Fid the positio of fial image formed. For les, v u f v 5 0 Hece it is object for mirror u 5 cm + v 5 0 v + 30 cm v 30 cm Now for secod time it agai passes through les u 5 cm v? ; f 0 cm v 5 0 v + 30 Hece fial image will form at a distace 30 cm from the les towards left. What should be the value of d so that image is formed o the object itself. For les : Case I : v 5 0 v + 30 cm If d 30, the object for mirror will be at pole ad its image will be formed there itself. Case II : If the rays strike the mirror ormally, they will retrace ad the image will be formed o the object itself d cm Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 37

38 3. Trasverse magificatio (m) Trasverse magificatio (m) of (of dimesio perpedicular to pricipal axis) is give by Ex.56 m v u If the les is thick or/ad the medium o both sides is differet, the we have to apply the formula give for refractio at spherical surfaces step by step. A exteded real object of size cm is placed perpedicular to the pricipal axis of a covergig les of focal legth 0 cm. The distace betwee the object ad the les is 30 cm. (i) Fid the lateral magificatio produced by the les. (ii) Fid the height of the image. (iii) Fid the chage i lateral magificatio, if the object is brought closer to the les by mm alog the pricipal axis. Usig ad v u f m u v we get m m f f + u...(a) ( 30) ve sig implies that the image is iverted. (ii) (iii) h h m h mh ( ) () 4 cm Differetiatig (A) we get f dm (f + u) du (0) ( 0) (0.) 00.0 Note that the method of differetial is valid oly whe chages are small. Aliter u (after displacig the object) ( ) 9.9 cm Applyig the formula m f f + u 0 m ( 9.9) chage i m 0.0. Sice i this method differetial is ot used, this method ca be used for ay chages, small or large. Q.3 A exteded real object is placed perpedicular to the pricipal axis of a cocave les of focal legth 0 cm, such that the image foud is half the size of object. (a) Fid the object distace from the les (b) Fid the image distace from the les ad draw the ray diagram (c) Fid the lateral magificatio if object i moved by mm alog the pricipal axis towards the les. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 38

39 3.3 Displacemet Method to fid Focal legth of Covergig Les : Fix a object of small height H ad a scree at a distace D from object (as show i figure). Move a covergig les from the object towards the scree. Let a sharp image forms o the scree whe the distace betwee the object ad the les is a. From les formula we have D a a f or a D a + f D 0...(A) This is quadratic equatio ad hece two values of a are possible. Call them a ad a. Thus a, ad a are the roots of the equatio. From the properties of roots of a quadratic equatio, a + a D a a f D Also (a a ) ( a + a) 4aa D 4fD d (suppose). d physically meas the separatio betwee the two positio of les. Q.3 Fid the relatio for the focal legth of les is terms of D ad d. Q.33 For what coditio, d 0, i.e. the two positio coicide Q.34 Roots of the eq (A) become imagiary if... Q.35 What type of image is formed o scree... (Real/virtual) Q.36 Prove that miimum distace betwee real poit object ad its image i case of covergig les is 4f. Q.37 If m ad m are the lateral magificatios i the two positio of les the show that m m. Q.38 If image legth are h ad h i the two cases, prove that h h H. 4. COMBINATION OF LENSES: Ex.57 The equivalet focal legth of thi leses i cotact is give by F f f f3 where, f f`, f 3 are focal legths of idividual leses. If two covergig leses are separated by a distace d ad the icidet light rays are parallel to the commo pricipal axis,the the combiatio behaves like a sigle les of focal legth give by the relatio d + F f f f f d F ad the positio of equivalet les is with respect to d les Fid the lateral magificatio produced by the combiatio of leses show i the figure. + f f f f + 0 v 0 0 v 0 0 f Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 39

40 0 0 cm Ex.58 0 m 0 Fid the focal legth of equivalet system. 3 f f f f f + f + f 3 00 f Q.39 Fid the equivalet focal legth of the system for paraxial rays parallel to axis. 5. COMBINATION OF LENS AND MIRROR : The combiatio of les ad mirror behaves like a mirror of focal legth f give by f F m Fl If leses are more the oe, f is give by Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 40

41 Ex.59 f F m For the followig figure f is give by f F m f l f + f Fid the positio of fial image formed. (The gap show i figure is of egligible width ) f eq f eq 0 cm + v 0 0 Hece image will be formed o the object itself Q.40 See the figure v 0 cm Fid the equivalet focal legth of the combiatio show i the figure ad positio of image. SOME INTERESTING FACT F CTS ABOUT LIGHT : () The Su Rises Before It Actually Rises Ad Sets After It Actually Sets : The atmosphere is less ad less dese as its height icrease, ad it is also kow that the idex of refractio decrease with a decrease i desity. So, there is a decrease of the idex of refractio with height.due to this the light rays bed as they move i the earth s atmosphere () The Su Is Oval Shaped At The Time Of Its Rise Ad Set : The rays divergig from the lower edge of the su have to cover a greater thickess of air tha the rays from the upper edge. Hece the former are refracted more tha the latter, ad so the vertical diameter of the su appears to be a little shorter tha the horizotal diameter which remais uchaged. (3) The Stars Twikle But Not The Plaets. The refractive idex of atmosphere fluctuates by a small amout due to various reasos. This causes slight variatio i bedig of light due to which the apparet positio of star also chages, producig the effect of twiklig. (4) Glass Is Trasparet, But Its Powder Is White : Whe powerded, light is reflected from the surface of iumerable small pieces of glass ad so the poweder appears white. Glass trasmits most of the icidet light ad reflects very little hece it appears trasparet. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 4

42 (5) Greased Or Oiled Paper Is Traspret, But Paper Is White : The rough surface of paper diffusely reflects icidet light ad so it appears white. Whe oiled or greased very little reflectio takes place ad most of the light is allowed to pass ad hece it appears trasparet. (6) A Exteded Water Tak Appears Shallow At The Far Ed : (7) A Test Tube Or A Smoked Ball Immersed I Water Appears Silvery White Whe Viewed From The Top : This is due to Total iteral reflectio (8) Ships Hag Iverted I The Air I Cold Coutries Ad Trees Hag Iverted Udergroud I Deserts: This is due to Total iteral reflectio Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phoe : , page 4