UNIT I LESSON - 1 CONTENTS

Transcription

1 Page UNIT I LESSON - CONTENTS. Aims and Ojectives.. Convex and Concave Lenses.a. Reraction through a thin lens.. Equivalent Focal Length o Two Thin Lenses Separated y a distance.. Aerrations in Lenses.. Introduction... Spherical aerration:...chromatic Aerration in a Lens:..3. Condition or Achromatism o Two Thin Lenses placed in contact..4.condition or Achromatism o Two Thin Lenses separated y a inite distance.3.coma.3..astigmatism and its minimization.3..4 Let us sum up.5 Check your Progress.6 Lesson end Activities.7 Points or Discussion.8 Reerences. Aims and Ojectives From this lesson you will understand aout the principle employed in convex and concave lenses and study aout dierent types o deects and to minimize them... Convex and Concave Lenses A convex lens is a transparent reracting medium ounded y two spherical suraces. The line joining the centres o curvature o the two suraces is called the principal axis. A section o the lens through its principal axis is called its principal section.

2 Page Principal Focus and Focal Planes AB represents the principal axis o a lens (Fig. ). Fig First Principal Focus (F ) : It is that point on the principal axis o the lens, the rays starting rom which (convex lens) or appear to converge at which (concave lens) ecome parallel to principal axis ater reraction rom the lens. The plane passing through F and perpendicular to the principal axis is called irst ocal plane. Second Principal Focus (F ) :Second principal ocus is that point on the principal axis at which the rays parallel to principal axis converge (convex lens) or appear to diverge (concave lens) ater reraction rom the lens. The plan passing through F, and normal to the principal axis is called the second ocal plane. The points F, and F, are called ocal points.

3 Page 3.a. Reraction through a thin lens Consider a thin lens o reractive index n placed in a medium o reractive index n l (Fig...) Let R, and R e the radii o the irst and second suraces respectively. Consider a point oject O placed on the principal axis. Let I' e the position o the image ormed y reraction at the irst surace. Fig.. Let V e the distance o image I' and u the distance o the oject rom the irst surace. Then ' R n n u n V n () Now I' acts as the virtual oject or the second surace. The inal image I is ormed at a distance V rom the second surace. Since the rays now pass rom a medium o reractive index n into a medium o reractive index n we have R n n R n n V n V n () Adding (I) and (), R R n n u n V n Dividing throughout y n R R n u V or R R n n u V (3) here n n n is the reractive index o the material o the lens w.r.t air.

4 Page 4 First and second principal Foci: The irst principal ocus (F ) is the position o an oject on the principal axis or which the image is at ininity. The distance o the irst ocus rom the lens is the irst ocal length ƒ (Fig..a). I V =, then u= ƒ and sustituting in Eq. (3), we get, n ( n ) R R R R (4) The second principal ocus (F ) is the position o the image on the principal axis or which the oject is al ininity (u =). The distance o the second ocus rom the lens is the second ocal length, ƒ [Fig..]. V = ƒ, when u =. Sustituting in Eq. (3), n ( n ) R R R R (5) Thus the irst and second ocal lengths o a lens are numerically equal when the lens is placed in a uniorm medium. The general convention is to call the second principal ocal length as the ocal length ƒ o the lens. u n R R This ormula is known as Lens makers ormula. (6) Power o a lens: Power o a lens is its aility to converge or diverge the rays o light. The power o a lens is measured y the reciprocal o its ocal length, P = l/ƒ. The unit o power is dioptre. One dioptre is the power o a lens whose ocal length is metre. Convex lenses have positive power and concave lenses, negative power. Focal length o comination o two thin lenses in contact: When two thin lenses o ocal lengths ƒ, and ƒ are placed in contact with each other, then the ocal length o the comination is given y Power o the comination = P = P + P.. Equivalent Focal Length o Two Thin Lenses Separated y a distance: Let L, and L, e two thin lenses o ocal lengths ƒ, and ƒ placed in air coaxially a distance a apart. Consider a ray PA incident on L parallel to the axis at a height h, aove the axis

5 Page 5 [Fig..]. This ray ater reraction through the irst lens is directed towards D which is the second principal ocus o L l. Fig.. Then, deviation produced y irst lens = h The emergent ray rom the irst lens strikes the lens L, at a height h,. The lens L deviates it urther through an angle. Finally the ray meets the axis at F which is the second principal ocus o the lens system. Deviation produced y the second lens = h PA and,f B are produced to cut at E. Then a single convex lens placed in the position E P and having ocal length P F is equivalent to the leas system. Thus P F = ƒ is the equivalent local length. Then, Deviation produced y the equivalent lens = h Now = + h h () h Now ah a h O B OK BK h a h h h Sustituting the value o h in Eq(), h h a

6 Page 6 or a () a (3) Here, a is known as the optical separation or optical interval etween the two lenses. Let us ind the position o the equivalent lens, i.e., the distance O P. The triangles BF,O, and E F P are similar. a h h a P F h h F oro P F F O h h Now, a a F O F P P O a a a a Let O P =-(P lies to the let o the lens L ). a a a (4) Similarly, consider a ray parallel to the axis incident rom the right hand side [Fig..(c)]. Then we can ind the position o F, the point where the ray intersects the principal axis ater reraction through the lens system. E P is the irst principal plane. P is the irst principal point o the lens system. The distance o the irst principal point rom the irst lens is =O P. Fig.(c)

7 Page 7 O P a a a The irst principal ocus F, is situated at a distance ƒ towards the let o the point P. I P and P are the powers o the component lenses and P the power o the comination, then, P=P +P -ap P. ABERRATIONS IN LENSES:.. Introduction: The deviations in the size, shape, position and colour in the actual images produced y a lens in comparison to the oject are called aerrations produced y a lens. Chromatic aerrations are distortions o the image due to the dispersion o light in the lenses o an optical system when white light is used. The deect o coloured image ormed y a lens with white light is called chromatic aerration. I monochromatic light is used, then such deects are automatically removed. Besides these deects, there are deects which are present even when monochromatic light is used. Such deects are called monochromatic aerrations. These aerrations are the result o (i) the large aperture o the optical system, (ii) the large angle sutended y the rays with the principal axis and (iii) the large size o the oject. As a result o these aerrations, (i) a point is not imaged as a point, (ii) a plane is not imaged as a plane and (iii) equidistant points are not imaged as equidistant points. Following are the monochromatic aerrations: (i) Spherical aerration, (ii) Astigmatism, (iii) Coma, (iv) Curvature o ield and (v) Distortion. Spherical Aerration in a Lens : This aerration is due to large aperture o the lenses. The lens o large aperture may e thought to e made up o lanes. Fig...

8 Page 8 The marginal and paraxial rays orm the images at dierent places. Fig..3. shows that a monochromatic point source S on the axis is imaged as S P, and S m. Here, S P, and S m are the images ormed y marginal and paraxial rays respectively. Thus the point oject is not imaged as a point. Similarly the ocus o marginal and paraxial rays do not coincide. The distance S P S m on the axis measures longitudinal spherical aerration.... Spherical aerration: The ailure o a lens to orm a point image o a point oject on the axis is called spherical aerration. Fig...(a) Fig...() For rays parallel to principal axis, the distance etween the oci o marginal and paraxial rays gives the extent o longitudinal spherical aerration. In Fig...(a) Fp and Fm are the ocii or the paraxial and the marginal rays respectively. Spherical aerration o a convergent lens is taken to e positive as the distance (ƒp ƒm), measured along the axis. The spherical aerration o a diverging lens is negative (Fig... ). Methods o minimizing Spherical Aerration:

9 Page 9 The ollowing methods are used to reduce spherical aerration. (i) By using stops: By using stops, we can reduce the lens aperture. We can use either paraxial or marginal rays. Here, circular discs, called the stops, are used to cut o the unwanted rays. It eliminates marginal rays or paraxial rays. But the use o stops reduce the intensity o the image and the resolving power o the instrument. (ii) By using the two lenses separated y a distance: When two convex lenses separated y a inite distance are used the spherical aerration is minimum when the distance etween the lenses is equal to the dierence in their ocal lengths. In this arrangement, the total deviation is equally shared y the two lenses, Hence the spherical aerration is minimum. (iii)by using a crossed lens: The radii o curvature R, R o a thin lens satisy the ollowing relation: n R R It, there ore, shows that, spherical aerration depends upon (i) the reractive index o the lens medium (n) and (ii) the shape actor, which is determined y the ratio =R /R. I the reractive index o material o the lens is.5, the spherical aerration will e minimum when=r /R = - /6, A convex lens whose radii o curvatures ear the said ratio is called as a crossed lens. It is essential to divide the deviation on two suraces equally. The axial and marginal rays o light come to ocus with minimum o spherical aerration. Condition or Minimum Spherical Aerration o Two Thin Lenses- Separated y a distance: Spherical aerration may e minimized y using, two Plano-Convex lenses separated y a distance equal to the dierence in their ocal lengths. Let two plano convex lenses L and L o ocal lengths ƒ and ƒ e placed coaxially separated y a distance a (Fig..3.(c)). Consider a ray OA, parallel to principal axis, incident on lens L at height h aove the principal axis.

10 Page Fig...(c) The deviation produced y the lens L is given y, h () The reracted ray AB is incident at B at a height h rom the axis on lens L. The deviation produced y the lens L is given y, h () The ray AB produced meets the axis at F which is the principal ocus o lens L. Hence CF =ƒ. For minimum spherical aerration, the deviation produced y oth the lenses should e equal, i.e., = h h h or h (3) From similar triangles ACF and BDF we get AC BD CF DF CF h or CF CD h a (4) Comparing Eqs (3) and (4), we get

11 Page or a a This is the condition or minimum spherical aerration or two lenses separated y a distance...3.chromatic Aerration in a Lens: The ocal length o a lens is given y n R R Since n changes with the colour o light, ƒ must e dierent or dierent colours. This change o ocal length with colour is responsile or chromatic aerration. It is classiied into two types: (a) Longitudinal chromatic aerration, () Lateral chromatic aerration. a) Longitudinal chromatic aerration: A eam o white light is incident on a convex lens parallel to the principal axis (Fig..3.). The dispersion o colours takes place due to prismatic action o the lens. Violet is deviated most and red the least. Red rays are rought to ocus at a point arther than the violet rays. Evidently ƒ r > ƒ v. The dierence ƒ r - ƒ v is a measure o the axial chromatic aerration o a lens or parallel rays. Fig...3 Expression or Longitudinal chromatic aerration The ocal length o a lens is given y n R R Let ƒ v, ƒ r and ƒ y e the ocal lengths o the lens or violet, red and yellow colours respectively. Also let n v, n r,and n y, e the respective reractive indices. Then

12 Page R R n v v..() R R n r r () R R n y y (3) sutracting Eq.() rom Eq.(), R R n n r v r v or R R n n n n y y r v r v v r Now y r v n n n =dispersive power o the material o the lens; y r v y v r (4) () Lateral chromatic aerration: Fig...3(a) shows a convex lens and an oject AB placed in ront o the lens. The lens orms the image o white oject AB as B v A v and B r A r in violet and red colours respectively. The images o other colours lie in etween the two. Evidently, the size o red image is greater than the size o violet image (B r A r > B v A v ). The dierence (B r A r - B v A v ) is a measure o lateral or transverse chromatic aerration. Fig..3(a)

13 Page 3 Chromatic aerration is eliminated y : (i) keeping two lenses in contact with each other and (ii) keeping two lenses out o contact. Achromatic Comination o Lenses: When two or more lenses are comined together in such a way that the comination is ree rom chromatic aerration, then such a comination is called achromatic comination o lenses. The minimization or removal o chromatic aerration is called achromatisation. Chromatic aerration cannot e removed completely. Usually, achromatism is achieved or two prominent colours...4. Condition or Achromatism o Two Thin Lenses placed in contact : The ocal length o a thin lens is given y n R R () Here, ƒ is the ocal length, n the reractive index, R and R are radii o curvature o the two suraces o the lens. Now we know that ƒvaries with n. Thereore, dierentiating Eq. (), d dn R R dn dn n n R R n d dn or n Here ω is the dispersive power o the material o the lens. () Let ƒ and ƒ e the ocal lengths o the two lenses in contact and ω and ω w their dispersive powers. I ƒ is the ocal length o the comination, then

14 Page 4 Dierentiating Eq.(3) d d d d d ut and d d d (3) I the comination is to e achromatic, ƒ should e the same or all colours or dƒ=. or.(4) In order, thereore, to design an achromatic doulet o ocal length ƒ, the ocal lengths o the constituent lenses must satisy Eq. (3) and Eq. (4). Knowing ω, ω and ƒ, the magnitudes o ƒ and ƒ can e ound y solving Eqs. (3) and (4). Since ω, and ω are positive, ƒ and ƒ must e o opposite signs. That is, i one lens is convex, the other should e concave. Since the achromatic doulet is to ehave as a converging lens, ƒ must e less than ƒ. Consequently, ω, < ω. The converging lens is, thereore, made o crown glass (smaller dispersive power) and the diverging lens, o lint glass (larger dispersive power).( Fig...4) Fig...4.

15 Page Condition or Achromatism o Two Thin Lenses separated y a inite distance: Let us consider two convex lenses o ocal lengths ƒ,ƒ separated y a distance a (Fig...5). Fig...5 The ocal length o the comination is a () Dierentiating Eq.(), d d a d d d Now d and d a d Since oth the lenses are or the same material ω =ω a d For an achromatic comination, the ocal length ƒ should not change with colour. :.dƒ=. Hence, a or a

16 Page 6 a () That is, the distance etween the two coaxial lenses must e equal to hal the sum o their ocal lengths..3.coma : When a lens is corrected or spherical aerration, it orms a point image o a point oject situated on the axis. But i the point oject is situated o the principal axis, the lens, even corrected or spherical aerration, orms a comet-like image in place o point image. This deect in the image is called coma. Consider an o axis point A in the oject (Fig..3) The rays leaving A and passing through the dierent zones o the lens such as,,33 are rought to ocus at dierent points B,B,B 3, gradually nearer to the lens. The radius o these circles go on increasing with increase in radius o zone. Thus the resultant image is comet like. Fig..3. Removal o coma. The comatic aerration may e eliminated as ollows:. By using a stop eore the lens and so making the outer zones ineective.. By properly choosing the radii o curvature or the lens suraces. For example, or an oject situated at ininity, the comatic aerration may e minimized y taking a lens o n =.5 and R k R 9 3. Ae sine condition. Ae showed that coma may e eliminated i each zone o the lens satisies the Ae sine condition

17 Page 7 n h sinθ = n h sin θ, Here, n and n are reractive indices o the oject and image regions respectively. h and h are the heights o the oject and the image. θ and θ are the angles which the incident and the conjugate emergent rays make with the axis. (Fig.l.4a) Fig.3a I this condition is satisied, the lateral magniication h n sin h n sin will e same or all the rays o light, irrespective o the angles θ and θ. Thereore, coma will e eliminated..3..astigmatism and its minimization: Consider a point B situated o the axis in a line oject which is vertically elow the axis o the lens. When the cone o rays rom B alls on ull circumerence o the lens, then ater reraction all the rays do not meet at a single point ( Fig.4.) a) The rays lying in the vertical plane BMN (called meridional plane) orm the image as a horizontal line P. () The rays lying in the horizontal plane BRS (called sagittal plane) orm the image as a vertical line S. The circle o least conusion C lies in etween P and S. The est image or the oject point is otained here. This deect is called astigmatism. The distance etween P and S is a measure o astigmatism and is called the astigmatic dierence.

18 Page 8 Fig.3. The astigmatic dierence in the concave lens is in opposite direction to that produced y a convex lens. Hence astigmatism may e reduced y suitale comination o concave and convex lenses. Such a comination o lenses is called anastigmatic comination. It is used in the construction o ojective lens in a photographic camera..4 Let us sum up In this lesson you learned aout the principle employed in convex and concave lenses and studied aout the dierent types o deects and how to minimize them..5 Check your Progress ) What is meant y principal ocus and ocal planes in convex and concave lenses ) Deine spherical and chromatic aerrations in lenses 3) What is ae sine condition in a convex lens 4) What is coma and astigmatism

19 Page 9.6 Lesson end Activities ) An achromatic telescope ojective o.5 m ocal length consists o two thin layers in contact with each other and their dispersive power are.5 and.7 respectively. Calculate their ocal lengths. ) Two thin converging lenses o ocal lengths, cm and cm are separated y a distance a. Calculate the eective ocal length o the comination..7 Points or Discussion. What do you mean y spherical and chromatic aerration o a lens? Explain how they are caused. How would you correct chromatic aerration in the case o lens system in contact?. Otain an expression or the dispersive power o a lens in the condition or or achromatism o a comination o two thin coaxial lenses (i) when in contact and (ii) when separated y a dierence..8 Reerences ) Optics and spectroscopy - A. Murugesan ) Geometrical optics Brijlal and Suramanyam

20 Page LESSON CONTENTS. Aims and Ojectives. Eye-pieces.. Hygen s Eye-piece.. Ramsden s Eye-piece..3 Ae s Homogeneous Oil Immersion Ojective. Dispersion y A Prism.. Reraction through A Prism.. Angular and Chromatic Dispersions..3 Cauchy s Dispersion Formula..4 Dispersive Power.3 Rainow.3. Primary Rainow.3. Secondary Rainow.3.3 General Discussion.4 Let us Sum UP.5 Check your Progress.6 Lesson end Activities.7 Points or Discussion.8 Reerences. Aims and Ojectives From this lesson you will learn aout the usage o convex and concave lenses in dierent optical instruments as eye pieces. Also you will study aout the light relection through a prism and also aout the angular and chromatic dispersions. Also you will learn aout the ormation o dierent types o rainows.

21 Page. EYE-PIECES : An eye-piece is a comination o lenses designed to magniy the image already ormed y the ojective o a telescope and microscope. An eyepiece consists o two planoconvex lenses. F is called the ield lens and E the eye lens (Fig..5). The ield lens has large aperture to increase the ield o view. The eye lens mainly magniies the image. To reduce the spherical aerration, the lenses taken are plano-convex lenses. Further the ocal lengths o the two lenses and their separation are selected in such a way as to minimize the chromatic and spherical aerrations. A comination o lenses is used in an eyepiece o a simple lens magniier or the ollowing reasons: (i) The ield o view is enlarged y using two or more lenses. (ii) The aerrations can e minimized. Fig Huygens' Eye-piece: Fig... Construction: It consists o two plano- con vex lenses o ocal lengths 3ƒ(ield lens) and ƒ (eye lens), placed a distance ƒ apart [Fig..). They are arranged with their convex aces towards the incident rays. The eye-piece satisies the ollowing conditions o minimum spherical and chromatic aerrations. (i) The distance etween the two lenses or minimum spherical aerration is given y a = ƒ l ƒ. In Huygen's eyepiece, a= 3 ƒ - ƒ = ƒ. Hence this eye-piece satisies the condition o minimum spherical aerration.

22 Page (ii) For chromatic aerration to e minimum a. In Huygens' eyepiece, 3 a aerration.. Hence this eyepiece satisies the condition o minimum chromatic Fig.. Working: An eye-piece orms the inal image at ininity. Thus the ield lens orms the image I the irst Field Lens ocal plane o eye lens, i.e., at a distance ƒ to the let o eye-lens. Now the distance etween the ield lens and eye-lens is ƒ. Thereore, the image I lies at a distance to the right o ield lens. The image I ormed y the ojective o microscope or telescope acts as the virtual oject or the ield lens. Thus we treat I as the virtual oject or the ield lens, and I as the image o I due to it (Fig...a) or ν=ƒ., F=3ƒ, u=? We have u F or u 3 u 3 i.e. I should e ormed at a distance 3/ ƒ rom the ield lens. Thereore the rays coming rom the ojective which converge towards I are ocussed y the ield lens at I. The rays starting rom I emerge rom the eye-lens as a parallel eam.

23 Page 3 Fig...(a) Cardinal Points o Huygens Eyepiece The equivalent ocal length F o this eyepiece is a F F The second principal point is at a distance rom the eye lens. a a 3 The irst principal point is a distance α rom the ield lens. a a The position o the principal points P and P and the principal oci F and F are shown in Fig... Since the system is in air, the nodal points coincide with the principal points.

24 Page 4 Fig......Ramsden's Eyepiece Construction: It consists o two plano convex lenses each o ocal length ƒ. The distance etween them is (/3) ƒ [Fig...]. For achromatism, the distance etween the two lenses should e a. But here a =(/3)ƒ Fig... Thus in this eyepiece the chromatic aerration is only partly reduced. Similarly, or minimum spherical aerration, a. Hence the spherical aerration is not at all reduced. This is a demerit o this eyepiece. Working: I is the image ormed y the ojective o the microscope or telescope. It serves as an oject or eyepiece. The eyepiece is adjusted such that the image I ormed

25 Page 5 y the ield lens lies in the irst ocal plane o the eyelens [Fig...a]. Then the eyepiece orms the inal image at ininity. Since the ocal length o the eye lens is ƒ and a = (/3) ƒ, I is at a distance ƒ/3 rom the ield lens. Now, the image I due to ojective serves as the oject or ield lens. I is the image o I due to ield lens. Or,,, 3 F V to ind u we have u or u or F u V u Fig...(a) Thus the eyepiece its so adjusted that the image (I ) ormed y the ojective o telescope or microscope lies at a distance ƒ/4 towards the let o ield lens. The crosswire is placed at I. I serves as the oject or ield lens and its image is ormed at I. Cardinal points: The ocal length F o the equivalent lens is F a F 3 3 a a

26 Page 6 a 3 3 a The positions o the principal points P l and P and the principal oci F and F are shown in Fig...(). Since the system is in air, the nodal points coincide with the principal points. Fig...(). Distance o the irst principal ocus rom the ield lens o the eyepiece = F L = F P - = 3ƒ/4 - ƒ/=ƒ/4. Similarly the distance o the second principal ocus rom the eye lens is L F = P F - = 3ƒ/4 - ƒ/=ƒ/4. Comparison o Eyepieces Huygens' eyepiece Ramsden's eyepiece The image o the oject ormed y the ojective alls in etween the two lenses. Thereore, no cross wires can e used. For this reason. it is called a negative eyepiece. It satisies the condition or minimum spherical aerration. It satisies the condition or achromatism. It is generally used or iological oservations where no measurements are required. The image o the oject ormed y the ojective lies in ront o the ield lens. Thereore, cross wires can e used. For this reason, it is called a positive eyepiece. It does not satisy the condition or minimum spherical aerration. It does not satisy the condition or achromatism. It is used with instruments meant or physical measurements.

27 Page 7..3 ABBE S HOMOGENEOUS OIL IMMERSION OBJECTIVE: In a microscope the ojective is a lens system corrected or chromatic and spherical aerrations. In a microscope designed to magniy 5 times or more, the ojective alone produces a magniication o 5 or more. The microscope ojective is a hemispherical lens L o radius R with its plane surace directed towards the oject A. The plane surace is in contact with cedar wood oil having the same reractive index as that o the lens. Under such conditions no reraction can take place except at the spherical surace o the R lens L. I the oject is placed at A at a distance rom C, then the image ormed at I at a distance R rom C (Fig...3). Then the image will e ree rom spherical aerration. The magniication produced is CI R AC R Fig...3 I the second lens L has the centre o its concave surace at I then the rays emerging rom the hemispherical lens L will all on its lower concave surace normally and will pass undeviated. Then the rays all on the convex surace o L. The radius o the convex surace o the lens L is so selected that I is one o the planatic points. The rays appear to diverge rom J" which is the other a planatic point. This gives added magniication without introducing spherical aerration. This property o the lens L holds true only or rays rom the point A and not or points adjacent to it.

28 Page 8 The oject is immersed in oil such as cedar wood oil and the hemispherical lens with its plane ace is also immersed in the oil. This is known as Ae's homogeneous oil immersion ojective... DISPERSION BY A PRISM A eam o white light, when it passes through a prism is split up into constituent colours and that is called dispersion o light. The image thus ormed on a screen is called a. spectrum. Fig. The spectrum consists o visile and invisile regions. In the visile region the order o the colours is rom violet to red. The principal colours are given y the word VIBGYOR (Violet, Indigo, Blue, Green, Yellow, Orange and Red). The deviation produced or the violet rays o light is maximum and or red rays o light it is minimum. Fig.. represents the dispersion o a white ray o light y a prism in the visile region. The region o the spectrum, o wavelengths shorter than violet is called ultra-violet and the region o wavelengths longer than red is called inra-red. In the present chapter, the discussion relates only to the visile region o the spectrum. The reractive index or the material o a prism (or a lens) is dierent or dierent wavelengths (or colours). The deviation and hence the reractive index is more or lue rays o light than the corresponding values or red rays o light. The deviation and the

29 Page 9 reractive Index o the yellow constituent are taken as the mean values. I the dispersion through a prism does not ollow the order given y VIBGYOR, it is said to e anomalous dispersion....refraction THROUGH A PRISM: The reractive index o the material o a prism is given y A D sin A sin where A is the angle o the prism and D is the angle o minimum deviation. For a small angled prism where and reer to the angle o the prism and the angle minimum deviation (or small values o, the angle is also small and the sines o the angles are taken equal to the angles). Fig....represents the angles o deviation, and, produced in the lue, mean yellow and red rays o light. r

30 Page 3 Fig... The deviations, and r can e written as: ( ) or mean yellow light. ( ) or lue light and r ( r ) or red light The dierence in deviation etween two colors is called angular dispersion. ( ) ( ) r r ( r ) dividing r ( r ) ( ) ( r ) ( ) where and r are the reractive indices or the lue and red rays o light and is the r d reractive index or the mean yellow rays o light. The expression is called the dispersive power o the material o the prism. It is constant or two colours (or wavelengths] chosen end is represented y ω,

31 Page 3 r The reciprocal o ω is called the Constringence. It is also customary to represent, and r where F, D and C which correspond to the Fraunhoer lines (dark lines) in the solar spectrum. The P, D and O lines lie in the lue, yellow and red regions o the spectrum and their wavelengths are 486Å, 5893 Å and 6563 Å respectively. (l Å =l Angstrom unit= -8 cm)....angular AND CHROMATIC DISPERSIONS: The reractive index o the material o a prism is given y A sin A sin where A is the angle o the prism and θ is the angle o minimum deviation. A A sin sin A cos sin A I dθ is the dierence in the angle o deviation etween two spectral lines o wavelengths d λ and λ +d λ, then is called the angular dispersion etween the wavelengths λ and d λ +d λ. Dierentiating equation with respect to on the let hand side and θ y right hand side. dividing y d λ A cos d A sin d

32 Page 3 A cos d d d A sin d A sin d d d A cos d A sin sin d A d d is called chromatic dispersion o the material o the prism. The angular dispersion o d the material o a prism depends on the angle o the prism and reractive index o the material o the prism. Using a spectrometer and the given prism, a graph is drawn etween and λ ( along the Y-axis and λ along the X-axis). The tangent to the curve at d any point measures the chromatic dispersion d Sustituting this value o d in equation (iv), d..3. CAUCHY'S DISPERSION FORMULA: or that particular wavelength. d can he calculated. d When an electromagnetic wave is incident on an atom or a molecule, the periodic electric orce o the wave sets the ound charges into viratory motion. The requency with which these charges are orced to virate is equal to the requency o the wave. The phase o this motion as compared to the impressed electric orce will depend on the impressed requency. It will vary with the dierence etween the impressed requency and the natural requency o the charges. Dispersion can e explained with the concept o secondary waves that are produced y the induced oscillations o the ound charges. When a eam o light propagates through a transparent medium (solid or liquid), the amount o lateral scattering is extremely small. The scattered waves travelling in a lateral direction produce destructive intererence. However, the secondary waves travelling in the same direction as the incident eam superimpose on one another. The resultant viration will depend on the phase dierence