Chapter Nine RAY OPTICS AND OPTICAL INSTRUMENTS

1 Chapter NineRAY OPTICSAND INTRODUCTIONN ature has endowed the human eye (retina) with the sensitivity to detectelectromagnetic waves within a small range of the electromagneticspectrum. Electromagnetic radiation belonging to this region of thespectrum (wavelength of about 400 nm to 750 nm) is called light. It ismainly through light and the sense of vision that we know and interpretthe world around are two things that we can intuitively mention about light fromcommon experience. First, that it travels with enormous speed and second,that it travels in a straight line. It took some time for people to realise thatthe speed of light is finite and measurable. Its presently accepted valuein vacuum is c = 108 m s 1. For many purposes, it sufficesto take c = 3 108 m s 1.

2 The speed of light in vacuum is the highestspeed attainable in intuitive notion that light travels in a straight line seems tocontradict what we have learnt in Chapter 8, that light is anelectromagnetic wave of wavelength belonging to the visible part of thespectrum. How to reconcile the two facts? The answer is that thewavelength of light is very small compared to the size of ordinary objectsthat we encounter commonly (generally of the order of a few cm or larger).In this situation, as you will learn in Chapter 10, a light wave can beconsidered to travel from one point to another, along a straight line joining2022-23 Physics310them. The path is called a ray of light, and a bundle of such raysconstitutes a beam of this Chapter , we consider the phenomena of reflection, refractionand dispersion of light, using the ray picture of light.

3 Using the basiclaws of reflection and refraction, we shall study the image formation byplane and spherical reflecting and refracting surfaces. We then go on todescribe the construction and working of some important opticalinstruments, including the human MODEL OF LIGHTN ewton s fundamental contributions to mathematics, mechanics, and gravitation often blindus to his deep experimental and theoretical study of light. He made pioneering contributionsin the field of OPTICS . He further developed the corpuscular model of light proposed byDescartes. It presumes that light energy is concentrated in tiny particles called further assumed that corpuscles of light were massless elastic particles. With hisunderstanding of mechanics, he could come up with a simple model of reflection andrefraction.

4 It is a common observation that a ball bouncing from a smooth plane surfaceobeys the laws of reflection. When this is an elastic collision, the magnitude of the velocityremains the same. As the surface is smooth, there is no force acting parallel to the surface,so the component of momentum in this direction also remains the same. Only the componentperpendicular to the surface, , the normal component of the momentum, gets reversedin reflection. Newton argued that smooth surfaces like mirrors reflect the corpuscles in asimilar order to explain the phenomena of refraction, Newton postulated that the speed ofthe corpuscles was greater in water or glass than in air. However, later on it was discoveredthat the speed of light is less in water or glass than in the field of OPTICS , Newton the experimenter, was greater than Newton the himself observed many phenomena, which were difficult to understand in terms ofparticle nature of light.

5 For example, the colours observed due to a thin film of oil on of partial reflection of light is yet another such example. Everyone who has lookedinto the water in a pond sees image of the face in it, but also sees the bottom of the argued that some of the corpuscles, which fall on the water, get reflected and someget transmitted. But what property could distinguish these two kinds of corpuscles? Newtonhad to postulate some kind of unpredictable, chance phenomenon, which decided whetheran individual corpuscle would be reflected or not. In explaining other phenomena, however,the corpuscles were presumed to behave as if they are identical. Such a dilemma does notoccur in the wave picture of light. An incoming wave can be divided into two weaker wavesat the boundary between air and REFLECTION OF LIGHT BY SPHERICAL MIRRORSWe are familiar with the laws of reflection.

6 The angle of reflection ( , theangle between reflected ray and the normal to the reflecting surface orthe mirror) equals the angle of incidence (angle between incident ray andthe normal). Also that the incident ray, reflected ray and the normal tothe reflecting surface at the point of incidence lie in the same plane(Fig. ). These laws are valid at each point on any reflecting surfacewhether plane or curved. However, we shall restrict our discussion to thespecial case of curved surfaces, that is, spherical surfaces. The normal in2022-23 Ray OPTICS andOptical Instruments311this case is to be taken as normal to the tangentto surface at the point of incidence. That is, thenormal is along the radius, the line joining thecentre of curvature of the mirror to the point have already studied that the geometriccentre of a spherical mirror is called its pole whilethat of a spherical lens is called its OPTICAL line joining the pole and the centre of curvatureof the spherical mirror is known as the principalaxis.

7 In the case of spherical lenses, the principalaxis is the line joining the OPTICAL centre with itsprincipal focus as you will see Sign conventionTo derive the relevant formulae for reflection by spherical mirrors andrefraction by spherical lenses, we must first adopt a sign convention formeasuring distances. In this book, we shall follow the Cartesian signconvention. According to thisconvention, all distances are measuredfrom the pole of the mirror or the opticalcentre of the lens. The distancesmeasured in the same direction as theincident light are taken as positive andthose measured in the directionopposite to the direction of incidentlight are taken as negative (Fig. ).The heights measured upwards withrespect to x-axis and normal to theprincipal axis (x-axis) of the mirror/lens are taken as positive (Fig.)

8 Theheights measured downwards aretaken as a common accepted convention, it turns out that a single formulafor spherical mirrors and a single formula for spherical lenses can handleall different Focal length of spherical mirrorsFigure shows what happens when a parallel beam of light is incidenton (a) a concave mirror, and (b) a convex mirror. We assume that the raysare paraxial, , they are incident at points close to the pole P of the mirrorand make small angles with the principal axis. The reflected rays convergeat a point F on the principal axis of a concave mirror [Fig. (a)].For a convex mirror, the reflected rays appear to diverge from a point Fon its principal axis [Fig. (b)]. The point F is called the principal focusof the mirror. If the parallel paraxial beam of light were incident, makingsome angle with the principal axis, the reflected rays would converge (orappear to diverge) from a point in a plane through F normal to the principalaxis.

9 This is called the focal plane of the mirror [Fig. (c)].FIGURE The incident ray, reflected rayand the normal to the reflecting surface liein the same The Cartesian Sign distance between the focus F and the pole P of the mirror is calledthe focal length of the mirror, denoted by f. We now show that f = R/2,where R is the radius of curvature of the mirror. The geometryof reflection of an incident ray is shown in Fig. C be the centre of curvature of the mirror. Consider aray parallel to the principal axis striking the mirror at M. ThenCM will be perpendicular to the mirror at M. Let be the angleof incidence, and MD be the perpendicular from M on theprincipal axis. Then, MCP = and MFP = 2 Now,tan =MDCD and tan 2 = MDFD( )For small , which is true for paraxial rays, tan ,tan 2 2.

10 Therefore, Eq. ( ) givesMDFD = 2 MDCDor, FD = CD2( )Now, for small , the point D is very close to the point , FD = f and CD = R. Equation ( ) then givesf = R/2( ) The mirror equationIf rays emanating from a point actually meet at another point afterreflection and/or refraction, that point is called the image of the firstpoint. The image is real if the rays actually converge to the point; it isFIGURE Focus of a concave and convex Geometry ofreflection of an incident ray on(a) concave spherical mirror,and (b) convex spherical OPTICS andOptical Instruments313virtual if the rays do not actually meet but appearto diverge from the point when producedbackwards. An image is thus a point-to-pointcorrespondence with the object establishedthrough reflection and/or principle, we can take any two raysemanating from a point on an object, trace theirpaths, find their point of intersection and thus,obtain the image of the point due to reflection at aspherical mirror.