Light and Reflection Notes (NSEJS Batch 2024) PDF

9 LIGHT For NSEJS BATCH 2024 OPTICS It is a branch of physics which deals with the study of light. It is mainly divided into three parts: 1. Geometrical Optics or Ray Optics: It deals with the reflection and refraction of light....

9 LIGHT For NSEJS BATCH 2024 OPTICS It is a branch of physics which deals with the study of light. It is mainly divided into three parts: 1. Geometrical Optics or Ray Optics: It deals with the reflection and refraction of light. 2. Wave or Physical Optics: It is concerned with nature of light and deals with interference, diffraction and polarization. 3. Quantum Optics: It deals with the interaction of light with the atomic entities of matter such as photo electric effect, atomic excitation etc. LIGHT Light is the invisible form of energy that causes the sensation of vision. Light waves are electromagnetic waves (a) Nature of Light: Theories about Nature of Light: (i) Particle Nature of Light (Newton’s corpuscular theory): According to Newton light travels in space with a great speed as a stream of very small particles called corpuscles. According to this theory reflection and refraction of light are explained while this theory was failed to explain interference of light and diffraction of light. So wave theory of light was discovered. (ii) Wave Nature of Light: Huygen consider the light remains in the form of mechanical rays and he consider a hypothetical medium like ether for propagation of light waves. Later on, Maxwell declared light waves as electromagnetic waves, so there is no need of medium for the propagation of these waves. They can travel in vacuum also. The speed of these waves in air or in vacuum is maximum i.e., 3 × 108 m/s. Photoelectric effect was not explained with the help of wave theory, so Plank gave a new theory which was known as quantum theory of light. This theory is failed to explain photo electric effect. (iii) Quantum Theory of Light: According to ‘Planck’ light travels in the form of energy packets or quantas of energy called photons. The rest mass of photon is zero. Each quanta carries energy E = hν. h → Planck’s constant = 6.6 × 10–34 J-s. ν → frequency of light PW OLYMPIAD WALLAH 1  NSEJS BATCH 2024 Some phenomenons like interference of light, diffraction of light are explained with the help of wave theory but wave theory was failed to explain the photo electric effect of light. It was explained with the help of quantum theory. So, light has dual nature. (iv) Dual Nature of Light: De Broglie explained the dual nature of light, i,e,wave nature and particle nature. (i) wave nature: Light is a electromagnetic waves it is transverse in nature and propagate in vacuum (ii) Particle or Photon Nature: With the help of this theory Einstein explained the photo electric effect. (b) Source of Light: A body which emits light or reflect the light falling on it in all possible direction is said to be the source of light. The source can be point one or an extended one. The sources of light are of two types: (i) Luminous Source: Any object which by itself emits light is called as a luminous source. e.g.: Sun and stars (natural luminous sources), electric lamps, candles and lanterns (artificial luminous sources). (ii) Non-luminous Source: Those objects which do not emit light but become visible only when light from luminous objects falls on them. They are called non-luminous sources. e.g.: Moon, planets (natural non- luminous sources), wood, table (artificial non-luminous sources). (c) Medium of Light: Substance through which light propagates or tends to propagate is called medium of light (i) Transparent Object: Bodies that allow light to pass through them i.e. transmit light through them, are called transparent bodies. e.g.: Glass, water , air etc, (ii) Translucent Object: Bodies that can transmit only a part of light through them are called translucent objects. e.g.: Frosted or ground glass, greased paper, paraffin wax (iii) Opaque Object: Bodies that do not allow light to pass through them at all are said to be opaque object. eg. Chair, desk etc Mind it: Depending on composition optical medium are divided into two type Homogeneous Medium: An optical medium which has a uniform composition throughout is called homogeneous medium. e.g., Vacuum, distilled water, pure alcohol, glass, plastics, diamond, etc. Heterogeneous Medium: An optical medium which has different composition at different points is called heterogeneous medium. eg. Air, muddy water, fog, mist, etc PW OLYMPIAD WALLAH 2  NSEJS BATCH 2024 (d) Behaviour of Light at the Interface of two Media: Fig. 1: Behaviour of light at the Interface of two media When light travelling in one medium falls on the surface of a second medium the following three effects may occur: (i) A part of the incident light is turned back into the first medium. This is called reflection of light. (ii) A part of the incident light is transmitted into the second medium along a changed direction. This is called refraction of light. (iii) The remaining third part of light energy is absorbed by the second medium. This is called absorption of light. (e) Characteristics of Light: Some common characteristics of light are given below: (i) Light has dual nature i.e both wave and particle, nature. (ii) Light is an electromagnetic wave. (iii) Light does not require material medium for its propagation i.e. light can travel through vacuum. (iv) The speed of light in free space (vacuum) is 3 × 108 m/s. Its speed is marginally less in air. Its speed decreases considerably in glass or water. (v) Light undergoes reflection from polished surfaces such as mirrors, etc. (vi) Light undergoes refraction when it goes from one medium to another. (f) Some Definition Related to the Light: (i) Ray of Light: The path along which light energy travels in a given direction is called ray of light. A ray of light is represented as a straight line. The arrow head on it gives the direction of light. (ii) Beam of Light: A collection of rays of light is called beam of light. However, if the number of rays is too small then such a collection of rays is called Pencil of light. Beam are classified as (I) Parallel Beam: When the rays of light travel parallel to each other, then the collection of such rays is called parallel beam of light. For example, sun rays entering into a room through a ventilator constitute a parallel beam. (II) Converging Beam: When beam is converging then source will be virtual (III) Diverging Beam: In a divergent beam of light, the light rays spread out from a point. PW OLYMPIAD WALLAH 3  NSEJS BATCH 2024 REFLECTION OF LIGHT (a) Definitions of Reflection: The phenomena of bouncing back of light in same medium after striking at the interface of two media is called reflection of light. (b) General Definitions about Reflection: (i) Mirror: A smooth polished surface from which regular reflection can take place is called mirror. MM’ is the mirror as shown in figure. Fig. 4: Reflection of Light (ii) Incident Ray: A ray of light which travels towards the mirror is called incident ray. Ray AB is incident ray in figure. (iii) Point of incidence: The point on the mirror, where an incident ray strikes is called point of incidence. ‘B’ is the point of incidence in figure. (iv) Reflected Ray: A ray of light which bounces off the surface of a mirror, is called reflected ray. BC is reflected ray in figure. (v) Normal: The perpendicular drawn at the point of incidence, to the surface of mirror is called normal. BN is the normal in figure. (vi) Angle of Incidence: The angle made by the incident ray with the normal is called angle of incidence. ∠ABN is the angle of incidence in figure. It is denoted by ∠i. (vii) Angle of Reflection: The angle made by the reflected ray with the normal is called angle of reflection. ∠CBN is the angle of reflection in figure. It is denoted by ∠r. (viii) Glance Angle of Incidence: The angle which the incident ray makes with the mirror is called glance angle of incidence. ∠ MBA is the glance angle of incidence in figure. (ix) Glance Angle of Reflection: The angle which the reflected ray makes with the mirror is called glance angle of reflection. ∠M’BC is the glance angle of reflection in figure. PW OLYMPIAD WALLAH 4  NSEJS BATCH 2024 Laws of Reflection The reflection of light from a surface obeys certain laws called laws of reflection. They are: Fig. 5: Laws of Reflection (i) Angle of Incidence is equal to the angle of reflection, i.e., ∠i = ∠r. (ii) Incident ray, reflected ray and normal to the reflecting surface always lie in the same plane. ∗ Important information: A ray of light striking the surface normally retraces its path. Explanation: When a ray of light strikes a surface normally, then angle of incidence is zero i.e.,∠i = 0. According to the law of reflection, ∠r = ∠i, ∠ r = 0 i.e. the reflected ray is also perpendicular to the surface. Thus, an incident ray normal to the surface (i.e. perpendicular to the surface) retraces its path as shown in figure. Fig. 6: Normal Incidence Laws of reflection are also obeyed when light is reflected from the spherical or curved surfaces as shown in figure (a) and (b) Fig. 7: Reflection from curved surface (d) Regular and Irregular Reflection: PW OLYMPIAD WALLAH 5  NSEJS BATCH 2024 (i) Regular reflection: The phenomenon due to which a parallel beam of light travelling through a certain medium, on striking some smooth polished surface, bounces off from it, as parallel beam, in some other fixed direction is called Regular reflection. Fig. 8: Regular reflection Regular reflection takes place from the objects like looking glass, still water, oil, highly polished metals, etc. Regular reflection is useful in the formation of images, e.g., we can see our face in a mirror only on account of regular reflection. However, it causes a very strong glare in our eyes. (ii) Irregular reflection or Diffused reflection: Fig. 9: Irregular or Diffused Reflection The phenomenon due to which a parallel beam of light, travelling through some medium, gets reflected in various possible directions, on striking some rough surface is called irregular reflection or diffused reflection. The reflection which takes places from ground, walls, trees, suspended particles in air, and a variety of other objects, which are not very smooth, is irregular reflection. Irregular reflection helps in spreading light energy over a vast region and also decreases its intensity. Thus, it helps in the general illumination of places and helps us to see things around us. Mind it: Laws of reflection are always valid no matter whether reflection is regular or irregular. (e) Rectilinear Propagation of Light: (i) Definition: In simplest terms, rectilinear propagation of light means that light energy travels in straight lines. (ii) Examples of rectilinear propagation of light in everyday life: (I) When the sunlight enters through a small hole in a dark room, it appears to travel in straight lines. (II) The light emitted by the head light of a scooter at night appears to travel in straight lines. (III) If we almost close our eyes and try to look towards a lighted bulb, it appears to give light in the form of straight lines, which travel in various direction. PW OLYMPIAD WALLAH 6  NSEJS BATCH 2024 (iii) Experiment to prove rectilinear propagation of light: Take three wooden upright A, B and C having a small hole in the middle, such that the holes are at the same height from the base. Arrange the uprights along the edge of a table, such that holes are in the same straight line. Place a lighted candle towards the upright A, such that it is facing the hole. Look through the hole of upright C. The candle flame is clearly visible. Fig. 10: Rectilinear Propagation of Light Now displace upright B, slightly towards right or left. It is seen that candle flame is no longer visible. This shows that light travels in straight lines. (f) Plane Mirror (i) Image: An optical image is a point where rays of light converge actually or appear to diverge. The image of an extended object is an assembly of image points corresponding to various points on the object. (I) Real image: If the rays of light after reflection (or refraction) converge actually at a point then the image formed is called real image. It can be seen as well as obtained on a screen placed at the position of the image. (II) Virtual image: If the rays of light don’t converge actually but appear to diverge from a point then the image formed is called virtual image. It cannot be taken on screen. Both the real and virtual image can be photographed. Real Image Virtual Image 1 A real image is formed when 1 A virtual image is formed two or more reflected when two or more rays appear to rays meet a point in be coming from a point behind front of the mirror the mirror. 2 A real image can be obtained 2 A virtual image cannot be on a screen. obtained on a screen. 3 A real image is inverted with 3 A virtual image is erect with respect to the object. respect to the object. (ii) Image formation by plane mirror: (I) Formation of image of a point object by a plane mirror: Consider a plane mirror XY. Let a point object O is placed in front of the mirror as shown in figure. A ray OA is incident on the plane mirror at right angle to the mirror (i.e. ∠i = 0). The reflection takes place at A and the reflected ray retraces its path along AO. (∴ ∠r = 0). Fig. 11: Image of a Point Object PW OLYMPIAD WALLAH 7  NSEJS BATCH 2024 Another ray starting from O incident at point B on the mirror and the reflected ray goes along BC such that ∠ i = ∠r. The reflected rays AO and BC never meet each other. When the reflected rays AO and BC are produced backward, they appear to be coming from point I. In other words, reflected rays appear to diverge from point I. So point I is the virtual image of a point object O. Since there is no actual meeting of rays at point I. The position of image I is as far behind the plane mirror as the position of the object O in front of the plane mirror. i.e. OA = IA (see in figure). (II) Formation of image of an extended object by the plane mirror: Fig. 12: Image of an extended object Consider an extended object OA (say a pin) placed in front of a plane mirror XY at O. Each point of the object (i.e., pin) acts like a point source of light. The virtual image of each point of the extended object is formed behind the plane mirror as shown in figure. IA' is the virtual image of an extended object OA. Fig. 13: Image of an extended object In ∆'s BAC and BA'C ∠i = ∠r, ∠ACB = ∠ A ' CB = 90 °, ∴ ∠ABC = ∠'BC Also BC is common ∴ ∆ ABC and ∆ A'BC are congruent by ASA So AC = A'C i.e. perpendicular distance of object from the mirror is equal to the perpendicular distance of image from the mirror In ∆’s OBA and IBA' ∠BOA = ∠BIA' = 900º ∠OBA = ∠IBA' and so ∠OAB = ∠IA'B. Further as AB = BA' so they are also congruent by ASA Thus OA = IA' i.e., Size of object = Size of image PW OLYMPIAD WALLAH 8  NSEJS BATCH 2024 (III) Lateral inversion: When we look through the plane mirror, we find that the right eye of the image of our face appears as the left eye and the left eye of the image appears as the right eye. In other words, the right side of the object appears as the left side of the image and vice versa. This effect is known as lateral inversion. (1) Definition: The exchange of the right and left sides of an object and its image is known as lateral inversion. (2) Demonstration of lateral inversion: Fig. 14: Lateral Inversion Write a letter on a card. Place it in front of a plane mirror. We find that letter appears as , i.e., right of letter. P appears as left side of the image of letter P as shown in figure (3) Cause of Lateral inversion: Lateral inversion is due to the fact that the image of points on the object which are at a lesser distance from the mirror are formed nearer in the mirror and for those points which are at some more distance will be formed at larger distance. So the image appears to be laterally inverted. (IV) Characteristics of the image formed by a plane mirror: 1. The image formed by a plane mirror is virtual. 2. The image formed by a plane mirror is erect. 3. The size of the image formed by a plane mirror is same as that of the size of the object. If object is 10 cm high, then the image of this object will also be 10 cm high. 4. The image formed by a plane mirror is at the same distance behind the mirror as the object is in front of it. Suppose, an object is placed at 5 cm in front of a plane mirror then its image will be at 5 cm behind the plane mirror. 5. The image formed by a plane mirror is laterally inverted, i.e., the right side of the object appears as the left side of its image and vice-versa. (V) Number of Images formed when the object is placed between Two Plane Mirrors: When two plane mirrors are placed facing each other at an angle θ and an object is placed between them, multiple images are formed as a result of multiple reflections. 360º If is even then the number of image formed, θ 360º =n −1. θ 360° If is odd then: θ Case 1: If the object lies symmetrically, then 360° n= − 1. θ Case 2: If the object lies asymmetrically, then 360° n=. θ 360° Case 3: If is equal to fraction then number of images = [n] i.e. only integer part. θ PW OLYMPIAD WALLAH 9  NSEJS BATCH 2024 SPHERICAL MIRROR A mirror whose reflecting surface is a part of a hollow sphere of glass is known as spherical mirror. For example, a dentist uses a curved mirror to examine the teeth closely, large curved mirrors are used in telescopes. These are of two types convex and concave. In concave mirror, reflecting surface is concave but in convex mirror, reflecting surface is convex. (a) Some terms Related to Spherical Mirror: (i) Pole: The central point of a mirror is called its pole. (ii) Centre of curvature: The centre of the sphere of which the mirror is a part is called centre of curvature. (iii) Radius of curvature: The radius of the sphere of which the mirror is a part is called radius of curvature. (iv) Principal axis: The straight line joining the pole and the centre of curvature is called the principal axis. (v) Aperture: The size of the mirror is called its aperture. (vi) Principal focus: Focus of concave mirror Focus of convex mirror A parallel beam of light after reflection A parallel beam of light after reflection from a concave mirror converges at a from a convex surface diverges and the point in front of the mirror. This point rays do not meet. However on (F) is the focus of a concave mirror and producing backward, the rays appear to it is real. meet at a point behind the mirror. This point is focus of the convex mirror and it is virtual. PW OLYMPIAD WALLAH 10  NSEJS BATCH 2024 (vii) Focal length: The distance between the pole and the focus is called the focal length. The focal length is half the radius of curvature. (viii) Focal plane: A plane passing through the principal focus and at right angles to the principal axis of a spherical mirror is called the focal plane. (b) Concave and Convex Mirror: Convex mirror is a spherical mirror, whose inner (cave type) surface is silvered and reflection takes place at the outer (convex) surface. Concave mirror is a spherical mirror, whose outer bulged surface is silvered and reflection takes place from the inner hollow (cave type) surface. (i) Rules for the formation of images by concave and convex mirrors: (I) A ray incident parallel to the principal axis actually passes (concave) or appears to pass (convex) through the focus. (II) A ray incident through the centre of curvature (C) falls normally and is reflected back along the same path. (III) A ray incident through the focus is reflected parallel to the principal axis. PW OLYMPIAD WALLAH 11  NSEJS BATCH 2024 (ii) Formation of images by convex mirror: (I) When the object is placed at infinity then image is formed at the focus. The image formed is virtual, erect and extremely diminished. (III) Uses of convex mirror: Convex mirror is used as rear view mirror in automobiles like cars, trucks and buses to see the traffic at the back side. It is also used in street lamps. (iii) Formation of images by concave mirror: (I) When the object is placed between the pole and the focus, then the image formed is virtual, erect and magnified. (II) When the object is placed at the focus then the image is formed at infinity. The image is extremely magnified. (III) When the object is placed between the focus and the centre of curvature then the image is formed beyond the centre of curvature. The image formed is real, inverted and bigger than the object. (IV) When the object is placed at the centre of curvature, then the image is formed at the centre of curvature. The image formed is real, inverted and equal to the size of the object. PW OLYMPIAD WALLAH 12  NSEJS BATCH 2024 (V) When the object is placed beyond the centre of curvature, then the image is formed between the focus and centre of curvature. The image formed is real, inverted and diminished. (VI) When the object is placed at infinity then the image is formed at the focus. The image formed is real, inverted and extremely diminished in size. Summary of the images formed by a concave mirror: Position of Object Position of Image Size of Image Nature of Image At infinity At focus F Highly diminished Real and inverted Beyond C Between F and C Diminished Real and inverted At C At C Same size Real and inverted Between F and C Beyond C Enlarged Real and inverted At F At infinity Highly enlarged Real and inverted Between P and F Behind the mirror Enlarged Virtual and erect (VII) Uses of concave mirror: (1) They are used as shaving mirrors. (2) They are used as reflectors in car head-lights, search lights, torches and table lamps. (3) They are used by doctors to concentrate light on body parts like ears and eyes which are to be examined. (4) Large concave mirrors are used in the field of solar energy to focus sun-rays on the objects to be heated. Solar cooker: When a parallel beam of sunlight falls on a concave mirror, this beam is brought to the focus of the mirror (see figure). As a result of this, the temperature of an object (say a container containing uncooked food) placed at the focus increases considerably. Hence the food in the container is cooked. Spherical Reflector type solar cooker PW OLYMPIAD WALLAH 13  NSEJS BATCH 2024 (c) Mirror Formula: (i) Sign convention for measuring distance in concave and convex mirror: (I) All distances are measured from the pole. (II) The incident ray is taken from left to right. (III) Distances measured in the same direction as that of the incident ray are taken to be +ve. (IV) Distances measured in a direction opposite to the incident ray are taken to be –ve. (V) Distances measured upwards and perpendicular to principal axis are taken +ve. (VI) Distances measured downwards and perpendicular to principal axis are taken –ve. Focal length of concave mirror is –ve  Focal length of convex mirror is +ve    ∴    For real image v is –ve  for virtual image v is +ve  IMPORTANT: These signs are according to the rectilinear co-ordinate system. (ii) Mirror formula: The mirror formula is a relation relating the object distance (u), the image distance (v) and the focal length (f) of a mirror. 1 1 1 The mirror formula is: + = u v f above equation is known as mirror formula and is valid for both concave and convex mirrors. However, the quantities must be substituted with proper signs. Proof of mirror formula: Consider case of concave mirror when the object AB is placed beyond C, the image A' B' will be formed between C and F. Fig. 19: Mirror Formula Since the ∆ DGF & ∆ A 'B'F are similar (by AA similarity) DG GF ∴ =... (i) A 'B' B'F Also ∆ ABC is similar to ∆ A 'B'C AB CB ∴ =... (ii) A 'B' B'C Now as ADGB is a rectangle so AB = DG PW OLYMPIAD WALLAH 14  NSEJS BATCH 2024 DG CB So from (ii) =... (iii) A 'B' B'C From equation (i) and (iii) GF CB =... (iv) B'F B'C Since aperture is very small as compared to radius of curvature, so G is very close to P  PC  ∴ GF PF = = as  PF   2  PF CB PF PB − PC From (iv) = ⇒ = ( Here CB = PB − PC, B'F = PC − PB ) PB'− PF, B'C = B'F B'C PB'− PF PC − PB' PF PB − 2PF ⇒ = PB'− PF 2PF − PB ' −f − u − 2(−f ) ⇒ = ( Here PF = −f , PB = −v ) − u, PB ' = − v − (−f ) 2(−f ) − (− v) −f − u + 2f = − v + f −2f + v ⇒ 2f 2 − vf = uv − 2vf − uf + 2f 2 ⇒ uf + vf = uv on dividing by uvf, we get uf vf vu 1 1 1 + = ⇒ += uvf uvf uvf u v f 1 1 1 = + f u v (iii) Relation between radius of curvature and focal length of spherical mirrors: (I) Concave Mirror: Two parallel rays AB and DE after striking mirror form image at F. Fig. 20 (a): Relation between R.O.C. and focal length of concave mirror Here, ∠1 = ∠2 (by laws of reflection) ∠1 = ∠3 (alternate angles) ∴ ∠2 = ∠3 So ∆BFC is isosceles ∴ BF = CF Since aperture of the mirror is small, ∴ BF = PF and so, CF = PF or F is the mid-point of PC 1 R ∴ PF = ⋅ PC = 2 2 R or f = 2 PW OLYMPIAD WALLAH 15  NSEJS BATCH 2024 (II) Convex Mirror: Two parallel rays AB&DE after striking the mirror form virtual image at F. Fig. 20 (b): Relation between R.O.C. and focal length We know ∠1 = ∠2 (by laws of reflection) ∠1 = ∠4 (corresponding angles) Also, ∠2 = ∠3 (vertically opposite angles) ∠3 = ∠4 So ∆BFC is isosceles ∆ & BF = FC But BF = PF (since aperture of mirror is small) ∴ PF = FC hence F is the mid point of PC PC R ∴ PF = P or f = 2 2 (iv) Power of mirror: A spherical mirror has infinite number of focus. 1 Optical power of a mirror (in Diopters) = − f (in metre) (v) Magnification for concave mirror: The linear magnification of a spherical mirror is the ratio of height of the image (h2) formed by the mirror to the height of the object (h1) i.e. Height of image h 2 Linear magnification, m = = Height of object h1 The linear magnification is a number that simply tells us how much taller the image is than the object. For example, if m = 1, it means that the image and the object are of the same height. Another formula for magnification is: v f m=− = u f −u The arbitrary minus sign given to linear magnification has nothing to do with the relative sizes of the object and the image but we can use it to tell whether the image is erect or inverted w.r.t. object. Mind it: Always draw a rough ray diagram while solving a numerical problem. Otherwise we will be confused as to which distance should be taken as +ve & which –ve. For virtual image: m is +ve [as virtual image is erect ∴ h2 is +ve as well as h1 is +ve] For real image: m is –ve [as real image is always inverted ∴ h2 is –ve while h1 is +ve] PW OLYMPIAD WALLAH 16  NSEJS BATCH 2024 (vi) Longitudinal magnification: Fig. 21: Longitudinal magnification dv − v 2 = du u 2 If a small object lies along principal axis, du may indicate the size of the object and dv the size of its dv image along principal axis. In this is called longitudinal magnification. Negative sign indicates du inversion of image irrespective of nature of image and nature of mirror. Mint it: if object lying along the principal axis is not of very small size, the longitudinal V2 − V1 magnification = (it will always be inverted) u 2 − u1 (vii) Velocity of image: (I) Object Moving Perpendicular to Principal Axis: h 2 −v = h1 u dh − v dh ⇒ = dt u dt −v or velocity of image = (velocity of object) u This discussion is for velocity w.r.t. to mirror and along the y-axis. (II) Object Moving Along Principal Axis: dv − v 2 du = 2 dt u dt dv du Where is the velocity of image along principal axis and is the velocity of object along dt dt principal axis. This discussion is for velocity w.r.t. to mirror and along the x-axis. (viii) Newton's formula: If x and y are the distances (along the principal axis) of the object and image respectively from the principal focus, then xy = f2, where f is the focal length. (ix) Paraxial rays: The rays of light which fall near the pole making small angles with the principal axis of the mirror (x) Marginal rays: The rays of light which are parallel and travel far away from the principal axis of the mirror. (xi) Spherical Aberration: It is the inability of a spherical mirror to bring all the rays of a beam of light to its focus is called spherical aberration. It can be eliminated by using a paraboloid mirror. PW OLYMPIAD WALLAH 17  NSEJS BATCH 2024 Solved Examples Example: If two plane mirror are inclined at 60 degree angle then find number of image formed by two plane mirror when a object is placed between them. 360 360 Solution: =n = = 6 θ 60 Number of images = n – 1 = 6 – 1 = 5 Example: How can we distinguish between a plane mirror, a concave mirror and a convex mirror without touching them? Solution: We can distinguish between them by bringing our face close to each of them. All of them will produce different types of image of our face. Example: The sun (diameter d) subtends an angle θ radians 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? Solution: Since the sun is very distant, u is very large and so (1/u) is practically zero. 1 1 So +0= v −f i.e. V= –f i.e., the image of sun will be formed at the focus and will be real, inverted and diminished. Now as the rays from the sun subtend an angle θ radians at the pole, then in accordance with figure. where x is the diameter of the image of sun x tan θ = f x = f tanθ if θ is very small, then x = fθ Example: A 2.0 cm long object is placed perpendicular to the principal axis of a concave mirror. The distance of the object from the mirror is 30 cm and its image is formed 60 cm from the mirror on the same side of the mirror as the object. Find the height of the image formed. Solution: u = –30 cm, v = –60 cm h v −60 ∴ m =2 = − = − =−2 h1 u −30 ⇒ h 2 =−2h1 =−2 × 2 =−4cm ∴Height of the image is 4 cm. It is inverted. PW OLYMPIAD WALLAH 18  NSEJS BATCH 2024 Example: A 1.2 cm long pin is placed perpendicular to the principal axis of a convex mirror of focal length 12 cm, at a distance of 8 cm from it. (a) Find the location of the image. (b) Find the height of the image. (c) Is the image erect or inverted? Solution: Here f is +ve so f = 12 cm Also, u = –8 cm. 1 1 1 Using, + = v u f 1 1 1 1 1 5 or = − = + = v f u 12 8 24 24 ∴v= cm = 4.8 cm. 5 Given, h1 = 1.2 cm h v v We know, 2 = − ⇒ h2 = – × h1 = 0.72cm h1 u u Image formed is erect. Example: Find the position, size and the nature of the image formed by a spherical mirror from the following data. u = –20 cm, f = –15 cm, ho = 1.0 cm 1 1 1 Solution: We have + = v u f 1 1 1 1 1 1 or =− = − = − v f u −15cm −20cm 60cm or v = – 60 cm. The image is formed at 60 cm from the mirror. Since the signs of u and v are same, the image is on the same side as the object (to the left of the mirror) and hence it is real. The magnification is h v −60cm m =i = − = − = −3 h0 u −20cm The minus sign shows that the image is inverted. Its size is 3.0 cm Example: A 2 cm high object is placed at a distance of 32 cm from a concave mirror. The image is real, inverted and 3 cm in size. Find the focal length of the mirror and the position of the image v h Solution: We have, m = − =i u h0 From the equation, hi = –3 cm and h0 =2 cm h −3cm m= i = = −1.5 h0 2cm v v or − =−1.5 or = 1.5 or v = – 48 cm u −32cm 1 1 1 1 1 −5 We have = + = + = f u v −32cm −48cm 96cm −96cm or f = = −19.2cm 5 So the focal length of the concave mirror is 19.2 cm and the image is formed 48 cm in front of it. PW OLYMPIAD WALLAH 19  NSEJS BATCH 2024 REFRACTION OF LIGHT When light travels in the same homogeneous medium, it travels along a straight path. However, when it passes from one transparent medium to another, the direction of its path changes at the interface of the two media. This is called refraction of light. The phenomenon of the change in the path of the light as it passes from one transparent medium to another is called refraction of light. The path along which the light travels in the first medium is called incident ray and that in the second medium is called refracted ray. The angles which the incident ray and the refracted ray make with the normal at the surface of separation are called angle of incidence (i) and angle of refraction (r) respectively. Fig. 22: Refraction of Light Showing different cases of refraction it is observed that: When a ray of light passes from an optically rarer medium to a denser medium, it bends towards the normal (∠r < ∠i ), as shown in figure (A). When a ray of light passes from an optically denser to a rarer medium, it bends away from the normal (∠r > ∠i) as shown in figure (B). A ray of light travelling along the normal passes undeflected, as shown in figure (C). Here ∠i = ∠r = 0° (a) Cause of Refraction: We come across many media like air, glass, water etc. A medium is a transparent material through which light is transmitted. Every transparent medium has a property known as optical density. The optical density of a transparent medium is closely related to the speed of light in the medium. If the optical density of a transparent medium is low, then the speed of light in that medium is high. Such a medium is known as optically rarer medium. Thus, optically rarer medium is that medium through which light travels fast. In other words, a medium in which speed of light is more is known as optically rarer medium. On the other hand, if the optical density of a transparent medium is high, then the speed of light in that medium is low. Such a medium is known as optically denser medium. Thus, optically denser medium is that medium through which light travels slow. In other words, a medium in which speed of light is less is known as optically denser medium. PW OLYMPIAD WALLAH 20  NSEJS BATCH 2024 Speed of light in air is more than the speed of light in water, so air is optically rarer medium as compared to the water. In other words, water is optically denser medium as compared to air. Similarly, speed of light in water is more than the speed of light in glass, so water is optically rarer medium as compared to the glass. In other words, glass is optically denser medium as compared to water. When light goes from air (optically rarer medium) to glass (optically denser medium) such that the light in air makes an angle with the normal to the interface separating air and glass, then it bends from its original direction of propagation. Similarly, if light goes from glass to air, again it bends from its original direction of propagation. The phenomena of bending of light from its path is known as refraction. We have seen that the speed of light in different media is different, so we can say that refraction of light takes place because the speed of light is different in different media. Thus, the cause of refraction can be summarised as follows: Mind it: Refraction is the deviation of light when it crosses the boundary between two different media (of different optical densities) and there is a change in both wavelength and speed of light. The frequency of the refracted ray remains unchanged. The intensity of the refracted ray is less than that of the incident ray. It is because there is partial reflection and absorption of light at the interface. (b) Effects of Refraction of Light: (i) A pencil appears bent and short in water: Fig. 23 (a): A pencil appears bent and short in water Consider a pencil PQ. Let AQ portion of the pencil be dipped in water as shown in figure. Rays of light from the tip (Q) of the pencil bend away from the normal as they go from water to air i.e. denser to rarer medium. These rays appear to come from a point B. Thus, the dipped portion of the pencil appears as AB. Hence a pencil appears bent and short when immersed in water. (ii) A water tank appears shallow i.e. less deep than its actual depth: Fig. 23 (b): A water tank appears shallow PW OLYMPIAD WALLAH 21  NSEJS BATCH 2024 Consider an object O say a stone lying on the bed of a water tank as shown in figure. A ray (OB) of light from the object suffers refraction at the free surface of water in the tank and bends away from the normal along BC. The refracted ray BC appears to come from point I which is above the object O. Thus, the bed of the tank appears at the level of point I. In other words, water tank appears shallow. (iii) Apparent shift in the position of the sun at sunrise and sunset: Due to the atmospheric refraction, the sun is visible before actual sunrise and after actual sunset. Fig. 23 (c): Apparent shift in the position of the sun at sunrise and sunset With altitude, the density and hence refractive index of air-layers decreases. The light rays starting from the sun S travel from rarer to denser layers. They bend more and more towards the normal. However, an observer sees an object in the direction of the rays reaching his eyes. So to an observer standing on the earth, the sun which is actually in a position below the horizon, appears in the position S’, above the horizon. The apparent shift in the position of the sun is by about 0.50. Thus the sun appears to rise early by about 2 minutes and for the same reason, it appears to set late by about 2 minutes. This increases the length of the day by about 4 minutes (iv) Twinkling of stars: On a clear night, you might have observed the twinkling of a star, which is due to an atmospheric refraction of star light. The density of the atmosphere, as we know goes on decreasing as the distance above the sea level increases. For the sake of simplicity, air can be supposed to be made up of a very large number of layers whose density decreases with the distance above the surface of the earth. Therefore, the light from a heavenly body, such as a star, goes on gradually bending towards normal as it travels through the earth’s atmosphere. As the object is always seen in the direction of the light reaching the observer’s eye, the star appears higher up in the sky than its actual position. Further, the densities of the various layers go on varying due to the convection currents set up in air by temperature differences. Thus, the refractive index of a layer of air at a particular level goes on changing Due to these variations in the refractive indices of the various layers of air, the light from a star passing through the atmospheric air changes its path from time to time and therefore, the amount of light reaching the eye is not always the same. This increase or decrease in the intensity of light reaching the eye results in the change in apparent position or twinkling of the star. PW OLYMPIAD WALLAH 22  NSEJS BATCH 2024 (c) Laws of Refraction: There are two laws of refraction: (i) The incident ray, the refracted ray and the normal at the point of incidence lie in the same plane. sin i (ii) = constant called refractive index denoted by ‘µ ’. sin r sin i The above law is called snell’s law (willibrod snell). Eg. = 1 µ2 sin r Here 1µ2 is called refractive index of 2nd medium w.r.t. 1st medium. Laws refraction as well as plane for. e.i surfaces of types both for valid are spherical refracting.surfaces } (d) Refractive Index: (i) Refractive Index in terms of Speed of Light: The refractive index of a medium may be defined in terms of the speed of light as follows: The refractive index of a medium for a light of given wavelength may be defined as the ratio of the speed of light in vacuum to its speed in that medium. Speed of light in vacuum Refractive index = Speed of light in medium c or µ= v Refractive index of a medium with respect to vacuum is also called absolute refractive index (ii) Refractive Index in terms of Wavelength: Since the frequency (υ) remains unchanged when light passes from one medium to another, therefore, c λ vac × υ λ vac µ= = = v λ med × υ λ med The refractive index of a medium may be defined as the ratio of wavelength of light in vacuum to its wavelength in that medium. (iii) Relative Refractive Index: The relative refractive index of medium 2 with respect to medium 1 is defined as the ratio of speed of light (v1) in the medium 1 to the speed of light (v2) in medium 2 and is denoted by 1µ2. v1 λ1 µ 2 Thus, 1 µ 2= = = v 2 λ 2 µ1 As refractive index is the ratio of two similar physical quantities, so it has no unit and dimension. (I) Factors on which the refractive index of a medium depends are: (1) Nature of the medium. (2) Wavelength of the light used. (3) Temperature. (4) Nature of the surrounding medium. It may be noted that refractive index is a characteristic of the pair of the media and also depends on the wavelength of light, but is independent of the angle of incidence. (II) Physical significance of refractive index: The refractive index of a medium gives the following two informations: PW OLYMPIAD WALLAH 23  NSEJS BATCH 2024 (1) The value of refractive index gives information about the direction of bending of refracted ray. It tells whether the ray will bend towards or away from the normal. (2) The refractive index of a medium is related to the speed of light. It is the ratio of the speed of light in vacuum to that in the given medium. For example, refractive index of glass is 3/2. This indicates that the ratio of the speed of light in glass to that in vacuum is 2: 3 or the speed of light in glass is two- third of its speed in vacuum. (iv) Refractive Index in terms of apparent depth and real depth: Whenever we observe the bottom of a swimming pool or a tank of clear water, we find that the bottom appears to be raised i.e. the apparent depth is less as compared to its real depth. The extent to which the bottom appears to be raised depends upon the value of refractive index of the refracting medium. Fig. 24: Refractive Index in terms of apparent depth and real depth In above fig. ∠PQN 2 = ∠i & ∠N1QR = ∠r sin i ∴ W µa = sin r sin r or a µ w = ………(1) sin i As ∠N1QR = ∠OP 'Q = ∠r (corresponding angles) OQ In ∆OP'Q sin r = sin ∠OP 'Q = ……. (2) P 'Q and ∠i =∠PQN 2 =∠QPO (alt. Int. ( ∠ s)) OQ ∴ In ∆QOP sin i = sin ∠OPQ =……(3) PQ So, from (1), (2) and (3) OQ / P 'Q PQ = a µw = …….(4) OQ / PQ P 'Q Nearly normal direction of viewing angle i is very small PQ ≅ PO and P 'Q ≅ P 'O ∴ from (4) PO a µw = P 'O Re al depth ⇒a µ w = Apparent depth PW OLYMPIAD WALLAH 24  NSEJS BATCH 2024 (e) Refraction Through Glass Slab: (i) Refraction through a rectangular glass slab and principle of reversibility of light: Consider a rectangular glass slab, as shown in figure. A ray AE is incident on the face PQ at an angle of incidence i. On entering the glass slab, it bends towards normal and travels along EF at an angle of refraction r. The refracted ray EF is incident on face SR at an angle of incidence r′. The emergent ray FD bends away from the normal at an angle of refraction e. Thus the emergent ray FD is parallel to the incident ray AE, but it has been laterally displaced with respect to the incident ray. There is shift in the path of light on emerging from a refracting medium with parallel faces. (ii) Lateral shift: Lateral shift is the perpendicular distance between the incident and emergent rays when light is incident obliquely on a refracting slab with parallel faces. (iii) Factors on which lateral shift depends are: (I) Emergent ray is parallel to the incident ray if there is same medium on both sides light is laterally t sin(i − r) stifled by, d = cos r if angle of incidence is very small then t(i − r)  r  1 d= = ti 1 −  = ti 1 −  1  i  µ (1) Lateral shift is directly proportional to the thickness of glass slab. (2) Lateral shift is directly proportional to the incident angle. (3) Lateral shift is directly proportional to the refractive index of glass slab. (4) Lateral shift is inversely proportional to the wavelength of incident light. (II) Emergent ray is not parallel to incident ray if the medium on both the sides of slab are different. (iv) Principle of reversibility of light: Fig. 25: Refraction through a rectangular glass slab If a plane mirror is placed in the path of emergent ray FD then the path of the emergent ray along FD is reversed back, it follows the same path along which it was incident i.e. the incidence ray becomes the emergent ray & emergent ray becomes the incident ray. It is known as principle of reversibility of light. PW OLYMPIAD WALLAH 25  NSEJS BATCH 2024 Case-I: For light going from air to glass at point E. ∠i = angle of incidence ∠r = angel of refraction sin i a µg = ………….(i) sin r (aµg = absolute refractive index of glass) Case-II: For light going from glass to air at point F sin r ' ∠r ' =angle of incidence  ⇒g µa = where   ∠r =∠r ' sin e  ∠e =angle of refraction  sin r ⇒g µa = (as ∠e =∠i) sin i 1 sin i ∴ = ………(2) g µa sin r ∴ From (1) & (2) ∠e =∠i , hence incident ray and emergent ray are parallel. 1 µa = ⇒ g µ a ×g µ a =1 g µa g (f) Refraction through Compound slab: (i) Compound slab: A compound slab is made of two or more media (say water and glass) bounded by parallel faces and is placed in air. A compound slab can be made by placing a glass tray completely filled with water on a glass slab. When an incident ray AB travelling in air (medium 1) strikes the water surface (medium 2) at B, it is refracted along BC. In figure ∠ABN = i (incident angle) and ∠N 'BC = r1 (angle of refraction) Fig. 26: Lateral shifting of light in compound slab Now the ray BC acts as an incident ray for the surface separating glass slab and water. So the incident ray BC after striking this surface at C is refracted along CD in glass (medium 3). ∠BCN1 = r1 , which is equal to angle of refraction, now acts as angle of incidence. ∠DCN1 ' = r2 = angle of refraction The ray CD acts as an incident ray for the surface separating glass slab and air. So the incident ray CD after striking this surface at D is refracted along DE in air. The rays DE and AB are parallel, so ∠N 2 ' = DE = ∠ABN = i. in this case, ∠CDN =i2 Mint it: Incident ray AB and emergent ray DE will be parallel PW OLYMPIAD WALLAH 26  NSEJS BATCH 2024 (ii) Relation of Refractive index of different media: The refraction of light occurs because light has different speed in different media. Speed of light is maximum in vacuum or air. It is less in any other medium. Denser is the medium lesser is the speed of light. Refractive index of a medium depends not only on its nature and physical conditions, but also on the colour or wavelength of light. It is more for violet light and less for red light (VIBGYOR). To find refractive index of two media w.r.t. each other when their refractive indices w.r.t. air are given. A ray of light AB refracts from different medium as shown in figure below. (i) For refractive index at interface A′ B′ sin x a µw = ………..(i) sin y (ii) For refractive index at interface C'D' sin y a µw = ………(2) sin z (iii) For refractive index at interface E'F' sin z g µa = …………(3) sin x Multiply (i), (ii) & (iii) a µ w ×w µ g ×g µ a =1 1 µg = a µ w ×g µ a w a µg  1  µg = …………(iv)  as = a µg    aµ w  g µa w  and on reciprocal a µw aµ w = ………..(v) a µg ∴ In general we can write as: 1 µ3 2µ 3 = 1 µ2 1 µ2 µ2 = 1 µ3 3 PW OLYMPIAD WALLAH 27  NSEJS BATCH 2024 (g) Total Internal Reflection: The phenomenon of reflection when a ray of light travelling from a denser to rarer medium is sent back to the same denser medium, provided when it strikes the interface of the denser and the rarer media at an angle greater than the critical angle, is called total internal reflection. When a ray of light falls on the interface separating denser and rarer medium, it is refracted as shown in figure. As the angle of incidence increases, the refracted ray bends towards the interface. At a particular angle of incidence, the, refracted light travels along the interface and the angle of refraction becomes 90º. The angle of incidence for which angle of refraction becomes 900 is called critical angle ic sin i c w µa = sin 90º 1 ⇒ sin i c = a µw Fig. 27: Ray diagram showing total internal reflection When the angle of incidence becomes greater than the critical angle, there is no refracted light and all the light is reflected in the denser medium. This phenomenon is known as total internal reflection. (i) Conditions for total Internal Reflection: (I) The light should travel from denser to rarer medium. (II) The angle of incidence must be greater than the critical angle for the given pair of media.  IMPORTANT NOTE: During total internal reflection of light, the whole incident light energy is reflected back to the parent optically denser medium. (1) Critical angle of a medium depends upon the wavelength of light. Critical angle ∝ wavelength: Greater the wavelength, greater will be the critical angle. Thus, critical angle of a medium will be maximum for red colour and minimum for violet colour. (2) Critical angle depends upon the nature of the pair of media. Greater the refractive index, lesser will be the critical angle. (3) Image formed due to total internal reflection is much brighter because total light is reflected back into the same medium and there is no loss in intensity of light. (ii) Some Phenomena due to total Internal Reflection: (I) Working of Porro Prism: A right angled isosceles prism called Porro-Prism can be used in periscope or binocular The refractive index of glass is 1.5 and the critical angle is equal to 41.8º. When the ray of light falls on the face of a right angled prism at angle greater than 41.8º, it will suffer total internal reflection. Right angled prisms used to bend the light through 90º and 180º are shown in figure (a) and (b) respectively. A right angled prism used to invert the image of an object without changing its size as shown in figure. PW OLYMPIAD WALLAH 28  NSEJS BATCH 2024 Fig. 28: Working of porro prism Additional Information: Mirrors can also be used for bending the rays of light. But the intensity of the beam reflected by mirrors is low because even a highly polished mirror does not reflect the whole light. On the other hand, in Porro-prism the whole light is reflected. Therefore, there is no loss in intensity of light and hence image is bright. (II) Sparking or brilliance of a diamond: The refractive index of diamond is 2.5 which gives, the critical angle as 24º. The faces of the diamond are cut in such a way that whenever light falls on any of the faces, the angle of incidence is greater than the critical angle i.e. 24º. So when light falls on the diamond, it suffers repeated total internal reflections. The light which finally emerges out from few places in certain directions makes the diamond sparkling. (III) Shining of air bubble in water: The critical angle for water-air interface is 48º 45′. When light propagating from water (denser medium) is incident on the surface of air bubble (rarer medium) at an angle greater than 480 45’, the total internal reflection takes place. Hence the air bubble in water shines brilliantly. Fig. 29: Shining of air bubble in water (IV) Mirage: Mirage is an optical illusion of water observed generally in deserts when the inverted image of an object (e.g. a tree) is observed along with the object itself on a hot day. Fig. 30: A mirage formation in deserts PW OLYMPIAD WALLAH 29  NSEJS BATCH 2024 Due to the heating of the surface of earth on a hot day, the density and hence the refractive index of the layers of air close to the surface of earth becomes less. The temperature of the atmosphere decreases with height from the surface of earth, so the value of density and hence the refractive index of the layers of air at higher altitude is more. The rays of light from distant objects (say a tree) reaches the surface of earth with an angle of incidence greater than the critical angle. Hence the incident light suffers total internal reflections as shown in the figure. When an observer sees the object as well as the image he gets the impression of water pool near the object. (1) The mirage formed in hot regions is called inferior mirage. (2) Superior mirage is formed in cold regions. This type of mirage is called looming. (V) Optical pipe and optical fibres: Optical fibre is extremely thin (radius of few microns) and long strand of very fine quality glass or quartz coated with a thin layer of material of refractive index less than the refractive index of the strand. (If refractive index of the core is say 1.7 then refractive index of the coating is 1.5). The coating or surrounding of optical strands is known as cladding. The sleeve containing a bundle of optical fibres is called a light pipe. When light falls at one end of the optical fibre, it gets total internally refracted into the fibre. The refracted ray of light falls on the interface separating fibre and coating at an angle which is greater than the critical angle. The total internal reflection takes place again and again as shown in figure below. The light travels the entire length of the fibre and arrives at the other end of the fibre without any loss in its intensity even if the fibre is rounded or curved. Uses of Optical Pipe: (1) Optical fibres are used to transmit light without any loss in its intensity over distances of several kilometer. (2) Optical fibres are used in the manufacture of medical instruments called endoscopies. Light pipe is inserted into the stomach of the human being. Light is sent through few optical fibres of the light pipe. The reflected light from the stomach is taken back through the remaining optical fibres of the same light pipe. This helps the doctors to see deeply into the human body. Hence the doctor can visually examine the stomach and intestines etc. of a patient. (3) They are used in telecommunication for transmitting signals. A single fibre is able to transmit multiple signals (say3000) simultaneously without interference, whereas the electric wire can preferably transmit one signal at a time. (4) Optical fibres are used to transmit the images of the objects. (5) Optical fibres are used to transmit electrical signals from one place to another. The electrical signals are converted into light by special devices called transducers, then these light signals are transmitted through optical fibres to distant places. PW OLYMPIAD WALLAH 30  NSEJS BATCH 2024 (h) Refraction at Spherical Surfaces: Fig. 32: Refraction through spherical surfaces Spherical refracting surface is a refracting medium whose curved surface is a part of sphere. For paraxial rays incident on a spherical surface separating two media. µ 2 µ1 µ 2 − µ1 − = (where light moves from the medium of R.I. to the medium of R.I µ2) v u R Solved Examples Example: Calculate the speed and wavelength of light (i) in glass and (ii) in air, when light waves of frequency 6 × 1014 Hz travel from air to glass of µ = 1.5. Solution: Here v = 6 ×1014 Hz, µ = 1.5 v a 3 × 108 (i) In glass speed of light v g = = = 2 × 108 m / s µ 1.5 vg 2 × 108 Wavelength of light, λ g = = = 3.3 × 10−7 m. ν 6 × 1014 (ii) In air speed of light v a = 3 × 108 m / s v a 3 × 108 Wavelength of light, λ a = = = 5 × 10−7 m ν 6 × 10 14 4 Example: The depth of water in a tank is 4 meter. If the refractive index of water is , by how much distance 3 does the bottom of tank appear to be raised? Solution: Actual depth, d = 4m 4 R.I. water, = µ = 3 Actual depth µ= Apparent depth d 4 Apparent depth= = = 3m µ 4/3 So, bottom of tank appear to be raised by 4 – 3 = 1 m PW OLYMPIAD WALLAH 31  NSEJS BATCH 2024 Example: There is a small air bubble inside a glass sphere (µ=1.5) of radius 10 cm. The bubble is 4.0cm below the surface and is viewed normally from the outside. Find the apparent depth of the bubble. Solution: The observer sees the image formed due to refraction at the spherical surface when the light from the bubble goes from the glass to the air. Here u = – 4.0 cm, R= –10 cm, µ1 = 1.5 and µ2 = 1 We have, µ 2 µ1 µ 2 − µ1 − = v u R 1 1.5 1 − 1.5 or − = v −4.0cm −10cm 1 0.5 1.5 1 2 − 15 40 or = − = = or, v =− = −3.07cm v 10 4.0 v 40 13 Thus, the bubble will appear 3.07 cm below the surface. PW OLYMPIAD WALLAH 32  NSEJS BATCH 2024 SPHERICAL LENSES A lens is a piece of transparent refracting material bounded by two spherical surfaces or one spherical and other plane surface. A lens is the most important optical component used in microscopes, telescopes, cameras, projectors etc. (a) Basically Lenses are of two Types: (i) Convex lens or converging lens (ii) Concave lens or diverging lens (i) Convex lens and its types: A lens which is thick at the centre and thin at the edges is called a convex lens. The most common form of a convex lens has both the surfaces bulging out at the middle. Some forms of convex lens are shown in the figure. Fig. 33: Different types of convex lens (ii) Concave lens and its types: A lens which is thin at the middle and thick at the edges is called a concave lens. The most common form of a concave lens has both the surfaces depressed inward at the middle. Some forms of concave lenses are shown in the figure. Fig. 34: Different types of concave lens (b) Definitions in Connection with Spherical Lens: (i) Centre of curvature (C): The centre of curvature of the surface of a lens is the centre of the sphere of which it forms a part, because a lens has two surfaces, so it has two centres of curvature. In figure (a) and (b) points, C1 and C2 are the centres of curvature. (ii) Radius of curvature (R): The radius of curvature of the surface of a lens is the radius of the sphere of which the surface forms a part. R1 and R2 in the figure (a) and (b) represents radius of curvature. PW OLYMPIAD WALLAH 33  NSEJS BATCH 2024 (iii) Principal axis (C1 C2): It is the line passing through the two centres of curvature (C1 and C2) of the lens. Fig. 35: Characteristics of convex and concave lenses (iv) Optical centre: If a ray of light is incident on a lens such that after refraction through the lens the emergent ray is parallel to the incident ray, then the point at which the refracted ray intersects, the principal axis is called the optical centre of the lens. In the figure O is the optical centre of the lens. It divides the thickness of the lens in the ratio of the radii of curvature of its two surfaces. If the radii of curvature of the two surfaces are equal, then the optical centre coincides with the geometric centre of the lens. Fig. 36: Ray Diagram showing optic centre For a ray passing through the optical centre, the incident and emergent rays are parallel. However, the emergent ray suffers some lateral displacement relative to the incident ray. The lateral displacement decreases with the decrease in thickness of the lens. Hence a ray passing through the optical centre of a thin lens does not suffer any lateral deviation, as shown in the figure (b) and (c) above. (v) Principal foci and focal length: (I) First principal focus and first focal length: It is a fixed point on the principal axis such that rays starting from this point (in convex lens) or appearing to go towards this point (concave lens), after refraction through the lens, become parallel to the principal axis. It is represented by F1 or f′. The plane passing through this point and perpendicular to the principal axis is called the first focal plane. The distance between first principal focus and the optical centre is called the first focal length. It is denoted by f1 or f′. Fig. 37 (a): Ray diagram showing First principal focus PW OLYMPIAD WALLAH 34  NSEJS BATCH 2024 (II) Second principal focus and second focal length: It is a fixed point on the principal axis such that the light rays incident parallel to the principal axis, after refraction through the lens, either converge to this point (in convex lens) or appear to diverge from this point (in concave lens). The plane passing through this point and perpendicular to principal axis is called the second focal plane. The distance between the second principal focus and the optical centre is called the second focal length. It is denoted by f2 or f. Fig. 37 (b): Ray diagram showing Second principal focus Generally, the focal length of a lens refers to its second focal length. It is obvious from the above figures, that the foci of a convex lens are real and those of a concave lens are virtual. Thus the focal length of a convex lens is taken positive and the focal length of a concave lens is taken negative. If the medium on both sides of a lens is same, then the numerical values of the first and second focal lengths are equal. Thus f = f′ (vi) Aperture: It is the diameter of the circular boundary of the lens. (c) Convex Lens: (i) Rules for image formation by Convex Lens: The position of the image formed by a convex lens can be found by considering two of the following rays (as explained below). (I) A ray of light coming parallel to principal axis, after refraction through the lens, passes through the principal focus (F) as shown in the figure. Convex Lens (II) A ray of light passing through the optical centre O of the lens goes straight without suffering any deviation as shown in the figure. Convex Lens PW OLYMPIAD WALLAH 35  NSEJS BATCH 2024 (III) A ray of light coming from the object and passing through the principal focus of the lens after refraction through the lens, becomes parallel to the principal axis. (ii) Image formed by Convex Lens: The position, size and nature of the image formed by a convex lens depends upon the distance of the object from the optical centre of the lens. For a thin convex lens, the various cases of image formation are explained below: (I) When object lies at infinity: When an object lies at infinity, the rays of light coming from the object may be regarded as a parallel beam of light. The ray of light BO passing through the optical centre O goes straight without any deviation. Another parallel ray AE coming from the object, after refraction, goes along EA′. Both the refracted rays meet at A′ in the focal plane of the lens. Hence, a real, inverted and highly diminished image is formed on the other side of the lens in its focal plane. (II) When object lies beyond 2F: When an object lies beyond 2F, its real, inverted and diminished image is formed between F and 2F on the other side of the lens as explained below: A ray of light AE coming parallel to the principal axis, after refraction, passes through the principal focus F and goes along EF. Another ray AO passing through the optical centre O goes straight without suffering any deviation. Both the refracted rays meet at A’. Hence a real, inverted and diminished image is formed between F and 2F on the other side of the convex lens. PW OLYMPIAD WALLAH 36  NSEJS BATCH 2024 (III) When object lies at 2F: When an object lies at 2F, its real, inverted image having same size as that of the object is formed on the other side of the convex lens as explained below: A ray of light AE coming parallel to the principal axis, after refraction, passes through the principal focus F and goes along EF. Another ray AO passing through the optical centre O goes straight without suffering any deviation. Both the refracted rays meet at A’. Hence a real, inverted image having the same size as that of the object is formed at 2F on the other side of the lens. (IV) When object lies between F and 2F: When an object lies between F and 2F in front of a convex lens, its real, inverted and magnified image is formed beyond 2F on the other side of the lens as explained below: A ray of light AE coming parallel to the principal axis, after refraction, passes through the principal focus F and goes along EF. Another ray of light AO passing through the optical centre goes straight without any deviation. Both these refracted rays meet at A’. Hence a real, inverted and magnified image is formed beyond 2F on the other side of the lens. (V) When object lies at F: When an object lies at the principal focus F of a convex lens, then its real, inverted and highly magnified image is formed at infinity on the other side of the lens as explained below: A ray of light AE coming parallel to the principal axis, after refraction, passes through the principal focus F and goes along EF. Another ray of light AO passing through the optical centre O goes straight without any deviation. Both these refracted rays are parallel to each other and meet at infinity. Hence a real, inverted, highly magnified image is formed at infinity on the other side of the lens. PW OLYMPIAD WALLAH 37  NSEJS BATCH 2024 (VI) When object lies between O and F: When an object lies between the optical centre O and the principal focus F of a convex lens, then its virtual, erect and magnified image is formed on the same side as that of the object as explained below: A ray of light AE coming parallel to the principal axis, after refraction, passes through the principal focus F and goes along EF. Another ray of light AO passing through the optical centre goes straight without any deviation. Both these refracted rays appears to meet at A′, when produced backward. Hence virtual, erect and enlarged image is obtained on the same side of the lens. The results of image formation by a convex lens are summerised in the table: Position of the object Position of the image Size of the image Nature of the image At infinity At the focus F Highly diminished Real and inverted Beyond 2F Between F and 2F Diminished Real and inverted At 2F At 2F Same size Real and inverted Between F and 2F Beyond 2F Magnified Real and inverted At F At infinity Highly magnified Real and inverted Between O and F On the side of the Magnified Virtual and erect object (d) Concave Lens: (i) Rules for image formation by Concave Lens: The position of the image formed by a concave lens can be found by considering following two rays coming from a point object (as explained below). (I) A ray of light coming parallel to the principal axis, after refraction, appears to pass through the principal focus F of the lens, when produced backward as shown in figure (a). (II) A ray of light passing through the optical centre O of the lens goes straight without suffering any deviation as shown in figure (b). (ii) Image formed by Concave Lens: The image formed by a concave lens is always virtual, erect and diminished and is formed between the optical centre O and the principal focus F of the lens. For a thin concave lens of small aperture, the cases of image formation are discussed below: PW OLYMPIAD WALLAH 38  NSEJS BATCH 2024 (I) When the object lies at infinity: When object lies at infinity in front of a concave lens, a virtual, erect, highly diminished image is formed at the principal focus F as explained below. The rays of light AE and BD coming parallel to the principal axis of the concave lens, after refraction, go along EG and DH respectively. When extended in the back direction, these refracted rays appear to pass through the principal focus F. Hence a virtual, erect and highly diminished image is formed at the principal focus F (II) When the object lies between 0 and ∞: When an object lies at any position between the optical centre O and infinity in front of a concave lens, the image formed is virtual, erect, diminished and is formed between the optical centre O and the principal focus F as explained below. A ray of light AE coming parallel to the principal axis, after refraction, goes along EG and appears to pass through principal focus when produced backward and another ray which is passing through the optical centre O goes straight without any deviation. Both these refracted rays appear to meet at A′. Hence, a virtual, erect, diminished image is formed between O and F. The summary of image formation by a concave lens for different positions of the object is given in table. Position of the object Position of the image Size of the image Nature of the image At infinity At F Highly diminished Virtual and erect Between Between O and F Diminished Virtual and erect O and ∞ PW OLYMPIAD WALLAH 39  NSEJS BATCH 2024 (e) Power of a Lens: It is the measure of deviation produce by a lens. It is defined as the reciprocal of its focal length in metres. Its unit is Diopter (D) (f should always be in metres). 1 100 Power (P) = or focal length f (in m) focal length f (in m) Power of a convex lens is +ve (As it has a real focus and its focal length measured is +ve.) Power of a concave lens is –ve (As it has a virtual focus and its focal length measured is –ve.) Mind it: If two thin lenses are placed in contact, the combination has a power equal to the algebraic sum of the powers of two lenses, P = P1 + P2 1 1 1 ⇒ = + Here, f1 and f2 are the focal length of lenses and f is focal length of combination of f f1 f 2 lenses. (f) Lens Formula: Relation between object distance u, image distance v and focal length f is: 1 1 1 − = v u f Proof of lens formula: Relation between object distance u, image distance v and focal length f is: Fig. 38: Lens formula Let an object A B be kept on one side of lens (between F1 and F2) then image A’ B’ is formed on other side of lens (beyond 2F2). Now obeying sign convention the object distance OA = –u, the image distance OA’ = v. and the focal length OF2 = f. A 'B' OA ' Since ∆OA’B’ and ∆OAB are similar ∴ =……(i) AB OA A 'B' F2 A ' Again ∆ COF2 and ∆B’F2 are similar ∴ = OC OF2 A 'B' F2 A ' But OC = AB ∴ = …….(ii) AB OF2 Hence from equation (i) and (ii) OA ' F2 A ' OA '− OF2 = = OA OF2 OF2 v v−f ⇒ = ⇒ vf = − uv + uf −u f On dividing each term by u v f, we get uf uv uf = + uvf uvf uvf 1 1 1 =− + u f v 1 1 1 or = − f v u PW OLYMPIAD WALLAH 40  NSEJS BATCH 2024 (g) Linear Magnification: Linear magnification (m) is defined as the ratio of the size of the image to the size of the object A 'B' h 2 height of image = m = = , AB h1 height of object v if m is + ve(image is virtual & erect) also m = u if mis – ve(imageisreal&inverted) Mind it: Lens maker formula: 1  1 1   µ lens  1 1  = (µ − 1)  − =  − 1  −  f  R1 R 2   µ medium   R1 R 2  (where µ is absolute refractive index of lens material) Spherical Aberration in Lenses: The inability of a lens to focus all the rays of light falling on it at a single point is known as spherical aberration in the lens. Intensity or brightness of the image is proportional to the square of the aperture of the lens, i.e, I ∞ A2. that is why, the brightness of the image produced by a lens which is half painted black reduces to half. However, size of image remains the same because every part of a lens forms a complete image of an object. If a lens is cut horizontally into two equal halves as shown, then intensity of transmitted light 1 becomes half and aperture of lens becomes of its initial value. 2 If a lens is cut vertically into two equal halves as shown, then intensity of transmitted light and aperture of the lens remains same. Minimum distance between a real object and real image formed by a convex lens of focal length f is 4f. A convex lens of refractive index µ2 behaves as a convex lens in a medium of refractive index µ1 (< µ 2 ) and diverging lens in a medium of refractive index µ1 (> µ 2 ) A concave lens of refractive index µ2 behaves as a concave lens in a medium of refractive index µ1 (> µ 2 ). PW OLYMPIAD WALLAH 41  NSEJS BATCH 2024 Solved Examples Example: An object is placed 12 cm away from the optical centre of a lens. Its image is formed exactly midway between the optical centre and the object: (i) Draw a ray diagram to show the image formed. (ii) Calculate the focal length of the lens used. Solution: (i) The ray diagram is shown below. The image is virtual, erect and a diminished image (ii) Using lens formula 1 1 1 − =, we have v u f 1 1 1 + = (−6) (−12) f 1 1 1 =− + f 6 12 −2 + 1 1 = = − 12 12 f = – 12 cm Example: Two thin convex lenses of focal lengths 10 cm and 20 cm are placed in contact. Find the effective power of the combination. Solution: P = P1 + P2 100 100 100 100 P= + = + = 10 + 5 = 15D f1 f2 10 20 Example: An illuminated object and a screen are placed 90 cm. apart. What is the focal length and nature of the lens required to produce a clear image on the screen, twice the size of the object? Solution: As the image is real, the lens must be a convex lens and it should be placed between the object and the screen. Let distance between the object & the convex lens be x then u = –x, v = 90 –x v Now m = = −2 (image is real) u 90 − x =−2 ⇒ x =30 −x ∴ u = –30 cm, v = +60 cm Now, 1 1 1 1 1 1 1 3 1 = − = − = + = = f v u 60 −30 60 30 60 20 ∴f = 20 cm PW OLYMPIAD WALLAH 42  NSEJS BATCH 2024 OPTICAL INSTRUMENTS AND HUMAN EYE Visual Angle, Magnifying Power of Optical Instruments: Visual Angle: The angle which an object subtends at our eye is called the 'visual angle'. The apparent size of an object as seen by our eye depends upon the visual angle. Larger the visual angle, bigger the apparent size of the object. Fig. 39: Visual Angle of Human Eye Magnifying Power: The purpose of microscopes and telescopes is to increase the visual angle. Therefore, the power of these instruments is measured by their power of increasing the visual angle. This is the ratio of the visual angle subtended by the image formed by the instrument at the eye to the visual angle subtended by the object at the unaided eye. (a) Human Eye: The human eye is one of the most sensitive sense organ of sight which enables us to see the wonderful world of light and colour around us. It is like a camera having a lens system and forming an inverted, real image on a light sensitive screen inside the eye. The structure and working of the eye is as follows: (i) Structure and Working of Human Eye: The human eye has the following parts: Fig. 40: Structure of Human Eye (I) Sclera: It is the outer part of eye which protects the eye. It is hard, opaque and white in colour. (II) Cornea: It is a transparent spherical membrane covering the front of the eye. (III) Iris: It is a coloured diaphragm between the cornea and lens. (IV) Pupil: It is a small hole in the iris. (V) Eye lens: It is a transparent lens made of jelly like material. (µ = 1.396) (VI) Ciliary muscles: These muscles hold the lens in position. (VII) Retina: It is a back surface of the eye. (VIII) Blind spot: It is a point at which the optic nerve leaves the eye. A