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RAY OPTICS 1. DEFINITION When a light ray strikes the surface separating two media, a part of it gets reflected, i. e., returns back in the initial medium, it is known as reflection. KN TERMS RELATED TO REFLECTION 2.1 Ray: Aray of light is the straight line path of transfer of light energy. Itis represented by a straight line an arrow - head indicating the direction of propagation. 2.2 Mirror : Itis a highly polished smooth surface from which most of the incident | Smooth side light gets reflected. It is represented by a line with hatches in the reverse surface. 2.3. Object: A Point from which incident ray actually diverge is called real object. Or point at which incident rays appear to converge real ue YS app 9g Object Image is called virtual object. B. object is defined on the basis of incident ray. c. Minimum two rays are required to show the position of object 2.4 Image: A Point at which reflected or refracted rays actually converge is calledreal image. Or point from which reflected or refracted rays appear to diverge is called virtual image. B. Minimum two reflected or re-fracted rays are required to determine the image position. 3. Laws of reflection: There are three laws of reflection : [i] The angle of incidence is equal to the angle of reflection. [ii] | Theincident ray, the normal and the reflected ray lie in the same plane. [iii] © There is a phase change of x radians when light wave is reflected by permeable denser RAY OPTICS medium surface but no phase change occurs if it is reflected by rarer medium surface. 4. Reflection by plane mirror : [a] The image formed by the plane mirror is always errect, of the same size and at the same distance as the object is. [b] To see the full image in a plane mirror, its length is just half the height of the man and it has to be kept in specific position. [c] When the plane mirror or any reflecting surface is turned through an angle @ and if incident ray remains stationary, then the reflected ray will turn through 20. [d] Image of an object formed by a plane mirror is perverted & if object is real then virtual image formed. [e] Image due to a plane mirror is as far behind the mirror as is the object infront of it. [f] The magnification produced by the plane mirror is 1 i.e. the size of the image is equal to the size of the object. [a] When the two plane mirrors are parallel to each other, the number of images is infinity. [h] When two plane mirror are held at angle 0 with their reflecting surfaces facing each other and an object is placed between them, images are formed by successive reflections. First of all we will calculate, n = - then — RAY OPTICS [i] If the angle between the two mirrors is 0, the deviation produced by successive reflections is 5=6, + 8, = 20-26. Note : When reflection takes place by a smooth surface, it is called regular reflection but when reflec- tion takes place by a rough surface, it is called diffused reflection. Ex.1 Find the minimum height of a mirror where one can see his full image. Sol. Let HC is the height of the person and E is the position of his eyes. Now applying laws of reflection, 1 1 we have, ML'= 2 EL rFkk MH'= 3 HE Now, H’L' = MM! = MH’ =L'M’ =HL te tue oe 2 | =HL-5 HL 2 So the required height of the mirror be half of the height of the person Ex.2 Find the minimum height of a mirror required to see the complete wall behind him. Sol. From, AAA’M and AMEE’ wehave AM. ME’ m= 2Me’ a og ee Again from, DEE’ M' and D M' B' B we have E B phe r 1 => M'B’ = 2 E'M’ 3 RAY OPTICS Now, MM’ = A'B’—A' M- MB’ B’B = AB - 2(ME’ + EM’) = AB-2 MM | => MM = 3 AB Thus, minimum height of the mirror be /13 of the will and the person must be in the middle of the mirror and the wall. Ex.3 Two plane mirrors are inclined at an angle 0. Aray of light is incident on one mirror at an angle of incidence i. The rayis reflected from this mirror, falls on the second mirror from where it is reflected parallel to the first mirror. What is the value of i, the angle of incidence in term of 0 ? Sol. ‘The situation is illustrated in figure. XAis the incident ray. BC is the final reflected ray. Itis given that BC is parallel to mirror M,. Look at the assignment of the angles carefully. Now N, is normal to mirror M,. Therfore B=6 Then from AOAB @+B+90°-ij= 180° or 0+6+90°-j=180° or i=26-90° Thus if the angle of incidence is i = 26 — 90° , then the final reflected ray will be parallel to the first mirror. Ex. 4 Two plane mirrors are placed at an angle 0 so that a ray parallel to one mirror gets reflected parallel to the second mirror after two consecutive reflections. The value of 0 will be- (1)30° (2) 60° (3) 75° (4) 90° Sol. As shown in figure, rayAB goes to mirrorM, , gets reflected and travels along BC and then gets reflected by M, and goes in CD direction. It the angle between M, and M, be a, then IN AOAB 4OBC and 4OCB are equal to a “3a = 180° a =60° Hence correct answer is (2) RAY OPTICS 1. TYPES OF CURVED MIRROR 1.1. acurved mirroris a smooth reflecting part ( in any shape) of a symmetrical cuved surface such as paraboloidal, ellipsoidal, cylindrical of spherical. Note: Here we will discuss only on spherical mirrors. 1.2 Concave Mirror : If the reflection takes place from the inner surface of a sphrerical mirror, then the mirror is called mirror. 1.3. Convex Mirror: if the outer surface of the spherical mirror acts as a reflector then the mirror is called convex mirror. 2. TERMS RELATED TO SPHERICAL MIRROR 2.1 Centre of Curvature (C) : It is the centre of sphere of which the mirror is a part. 2.2 Radius of Curvature (R) : It is the radius of the sphere of which the mirror is a part. 2.3 Pole (P) : Itis the geometrical centre of the spherical reflecting surface. 2.4 Principal Axis : It is the straight line joining the curvature to the pole. 2.5 Focus (F) : When a narrow beam of rays of light, parallel to the principal axis and close to it ( know as paraxial rays), is incident on the surface of a mirror, the reflected beam is found to converge ( concave mirror) or appear to diverge ( convex mirror) from a point principal axis. This point is called focus. 2.6 Focal Length (F) : It is the distance the pole and the principal focus. For spherical mirrors, f= R/2 3. REFLECTION THROUGH CONCAVE MIRROR : F —» Principal focus P —> Pole of mirror C > Centre of curvature. PC = Radius of curvature. PF = Focal length. When a narrow beam of light travelling parallel to the principal axis is incident on the reflecting surface of the concave mirror, the beam after reflection converge at a point on the principal axis. RULES FOR RAY DIAGRAMS : 34 When a ray falls in the direction of centre of curvature of mirror then it reflects back along the same path. 3.2. Aray, parallel to the principal axis will after reflection, pass through the focus. 3.3. Aray, passing through the focus is reflected parallel to the principal axis. RAY OPTICS 3.4 Image formed by the concave mirror: Real, inverted & diminished Between infinity & Centre of Curvature At centre of curvature | Real, inverted and of the same size Between Focus & Behind the mirror Erect virtual & Pole enlarged RAY OPTICS 4. REFLECTION THROUGH CONVEX MIRROR : When a narrow beam of light travelling parallel to the principal axis is incident on the reflecting surface of the mirror, the beam after reflection appear to diverge from a point on the principal axis. 3 Note : [i] When a ray incident on convex mirror in the direction of centre of curvature after refelection comes back along the same path. [ii] When a ray incidet on convex mirror parallel to the principal axis, after reflection, appears to come from the focus. [iii] | Aray appearing to pass through the focus is reflected parallel to the principal axis. Image formed by convex mirror : A convex mirror forms only virtual images for all positions of the real object. The image is always virtual, erect, smaller than the object and is located between the pole & the focus. The image becomes smaller and moves closer to the focus as the object is moved away from the mirror. 5. SIGN CONVENTION : (1) Alldistances are measured from the pole. (2) The distance measured along the direction of propagation of light are taken as positive and the direction opposite to the propagation of light is taken as negative. (3) The distance(heights) measured above the principal axis i.e. along postive Y axis, are taken as positive while distances below the principal axis i.e. along negative Y axis are taken as. negative. 6. MIRROR FORMULAE : 6.1 If an object is placed at a distance u from the pole of a mirror and its images is formed ata distance v (from the pole) then In this formula to calculate unknown, known quantities are substituted with proper sign. 6.2 If a thin object linear size O situated vertically on the axis of a mirror at a distance u from the pole and its image of size | is formed at a distance v (from the pole). magnification (transverse) is defined as RAY OPTICS (+ve Erect image / | ‘i (-ve inverted image ¥% n=|3| --| | (|m| >1 large image ¥% (\m| < 1 Small image + Here-ve magnification implies that image is inverted with respect to object whiletive magnifica- tion means that image is erect with respect to object 6.3 Other formulae of magnification m= a m= i 6.4 The power of a mirror is defined as 1 100 ~~ #in m) ~~ Fin cm) The unit of power is diopter. P 6.5 The focal-length of a spherical mirror of radius R is given by 0) In sign convention, f (or R) is negative for concave or converging mirror and positive for convex or diveging mirror. 6.6 Newton'sformula f?=x, x, f= focal length of mirror | x, = Postion of object with respect to focus | x, = Postion of image with respect to focus. SPECIAL NOTE A For real extended object if the image formed by a single mirror is erect it is always virtual (i.e. m is +ve) and in this situation if the size of image is. So by observing the size of erect image in a mirror we can dicide the nature of the mirror i.e. whether it is convex concave or plane. 8 RAY OPTICS B. For real extended object if the image formed by a single mirror is inverted it is always real (i.e., m is —ve) and the mirror is concave. In this situation if the size of image is Smaller than object is ee Larger than object is between the «> and C and} Edual to object, Object is at C between C and F and Image image between Fandc | andimage between C & <0 c. As every part of a mirror forms complete image, if a part of mirror (say half) is obstructed (say convered with black paper) full image will be formed but intensity will be reduced. D. If an object moved at constant speed towards a concave mirror from infinity to focus, the image will move slower in the beginning and faster in the last, away from the mirror. This is because in the time object moves from = to C the image will move from F to C and when object moves from C to F the image will move from C to o .At C the speed of object and image will be equal. 2 v= 45 dv, Where v, velocity of image v 2 v= -(¢) Vo v, velocity of object Ex.1. Aconcave mirror of focal length f produces a real image n time the size of the object. What is the distance of the object from the mirror. f Sol, m=-n m= —— f-u => nf+nu =-f ge SD n RAY OPTICS Ex.2 The focal length of a concave mirror is 30 cm. Find the position of the object in front of the mirror, so that the image is three times the size of the object. Sol. Here image can be real or virtual. If the image is real f=-30, u=?, m=-3 f -30 ™-F9 = "mod u=-40 cm. If the image is virtual f 30 m=Fy => a = u=-20 cm. Ex.3 The sun (diameter D) subtands an angle 0) radian at the pole of a concave mirror of focal length f. What is the diameter of the image of the sun formed by the mirror. 1 Sol. Since the sun is very distant, u is very large and so = is practically zero 40 Met u vou f 1 1 vf vent The image of sun will be formed at the focus and will be real, inverted and diminished A'B' = height of image Arc A'B' d “Radius FP =8= 7 d=19 Ex.4 Abeam of light converges towards a point O, behind a convex mirror of focal length 20 cm. Find the nature and position of image if the point O is - (1) 10 cm behind the mirror (2) 30 cm behind the mirror Sol. Here objectis virtual (1) u=+10cem 0 f=+ 20cm uf 10x20 Te > Y= 40-20 =-20 cm. ifi | —_ v I 20 —_ oO magnification m=— | 7 m=-|75 =+ (2) u=+ 30cm, f= 20cm uf 30 x 20 ME ade = 30-20 v=+60 cm - # | : m=— 30 =-2 10 1. INTRODUCTION: The bending of the light ray from its path in passing from one medium to the other medium is called refraction of light. lf the refracted ray bends towards the normal relative to the incident ray, then the second me- dium is said to be denser than the first medium. But if the refracted ray bends away from the normal, then the second medium is said to be rarer than the first medium. Refraction v v AtPlane Surface At Spherical Surface AtAny Other Surface + + + t ‘ ByPlane BySlab By Prism By Spherical Surface By Lences 2. REFRACTIVE INDEX: [a] Refractive index of second medium w.r.t. first medium uw, _c/v, _ v, _ Velocityof light in first medium u, ec/v, Vv, Velocity of light in second medium [b] Absolute refractive index of medium (n or p) : bly = Velocity oflightinvaccum c_— Sini _ Velocity oflightinmedium v Sinr — {c] Refractive index is the relative property of two media. If the first medium carrying the incident Sini ray is a vaccum, then the ratio Sinr is called the ‘absolute refractive index of the second medium’. The relative refractive index of any two media is equal to the ratio of their absolute refractive indices. Therefore, if the absolute refractive indices of media 1& 2 ben, &n, respec- tively, then the refractive index of medium 2 with respect to medium 1 is- nz — Sini n, =n. = = re? "na, Sinr n, sini =n, sinr 41 RAY OPTICS | a [e] For three medium 1, 2, 3 due to successive refraction. 4M * aM; * 3n, = 1 n n. n. a Mes ny Ag Ag =1 [f] For two medium, n, & n, are refractive indices with respect to vaccum, the incident and emergent rays are parallel then n, Sing, = Ny sings. 3. LAW OF REFRACTION : [a] Snell's Law : For any two media and for light of a given wave length, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant. Sini Sine COnstant Where i= incidence angle r= refraction angle. [b] Theincident ray, the refracted ray and the normal at the incident point all lie in the same plane. [c] Frequency (and hence colour) and phase do not change ( while wavelength and velocity changes) 4. APPLICATION OF SNELL’S LAW : [a] When light passes from rarer to denser medium it bends to- ward the normal. Using Snell's Law u, sin0, = 1, sind, sin 0, _ Hy sin8, by Thus If p>, then 6, < 0, [b] | When light passes from denser to rarer medium it bends away from the normal. From Snell's law sin 4, _ By sm0, My, Thus If 5
0, 12 RAY OPTICS (c) | Whenlight propagates through a series of layers of different medium, then according to Snell's law u, SINO, =, SINB,= w, SINB, = .......... = constant (d) Conditions of no refraction {i) If light is incident normally on a boundary i.e., 2i=0° Then from Snell's law yu, sin 0 =u, sin r, by => sinr=Qie. 2r=Oie., Be light passes undeviated from the boundary. (so boundary will be invisible) (ii) If the refractive indices of two media are equali.e., if, Hy = Be = Then from Snell's law uw, sini=psinr (—sqeeseeeee Hi =H => 4i=Zrie, ray passes undeviated from the boundary with i= p,, Le., when the object is observed from a denser medium, itappears to be farther awy from the interface, 1.6. dap > dc (ii) If p< jy, 6., when the objectis observed from a rarer medium, it appears to be closer to the interface, i.e.d,
