8-OPTICS-01-THEORY

OPTICS 1. Optics is the branch of Physics, which deals with those electromagnetic radiations that produce sensation of vision. 2. To make the study of optics easy, it is divided into two parts. (a) Ray Optics (b) Wave Optics (a) The phenomena of rectilinear propagation of light, reflection and refraction, is studied in this section. (b) The wave behavior of light (like diffraction and interference) is studied in this section. REFLECTION OF LIGHT when light falls on the surface of a material it is either re-emitted without change in frequency or is absorbed in the material and turned into heat. When the re-emitted light is returned into the same medium from which it come, it is called reflected light and the process is known reflection. Law of Reflection : Note : To apply laws of reflection draw tangent (T) and Normal (N) at the point of reflection (P) 1st Law : Incident ray (I), normal (N) and reflected ray (R) are in same plane. 2nd Law : Angle of incidence = Angle of reflection Angle of deviation : The angle between the direction of incident and reflected (refracted) light ray is known as angle of deviation. angle of deviation Assume then Normal incidence : Grazing Incidence : and IMAGE FORMATION OF POINT OBJECT BY PLANE MIRROR 1. Point of intersection of incident light ray is known as object. 2. The point of intersection of reflected rays or refracted ray is known as image. Note : The object and image may be real or virtual 3. For real object the image formed by plane mirror is virtual. 4. For virtual object image formed by plane mirror is real. Field of view : Region in which diverging rays from object or image are actually present is known as field of view. Case 1 : Field of view for real object Case 2 : Field of view for virtual object Image formation of extended object by plane mirror : Case 1: Case 2 : Properties of image formed by a plane mirror of an extended object (i) size of the object = size of the image (ii) image is laterally inverted. Illustration : A man is standing at distance x from plane mirror in front of him. He wants to see the entire wall in mirror which is at distance y behind the man. Find the size of mirror ? Solution : From and size of mirror, Note : If Rotation of plane mirror : 1. If a plane mirror is rotated through an angle  about an axis in the plane of mirror then reflected ray, image and spot rotated through an angle 2 in the same sense. 2. If plane mirror is rotated through about an axis perpendicular to plane of mirror then reflected ray image, spot do not rotate. Velocity of image formed by a plane mirror : x co-ordinate of object relative to mirror x co-ordinate of image relative to mirror Here, differentiating, velocity of image relative to mirror = – velocity of object relative to mirror. Illustration : Find velocity of image when object and mirror both are moving towards each other with velocity 2 m/s and 3 ms-1. How they are moving ? Solution : Here Combination of plane mirror : 1. To find net deviation produced by combination of plane mirror add deviation produced by each mirror. While adding the deviation ensure that they must be in same direction either clockwise or anticlockwise. Illustration : Two plane mirrors are inclined at 30º as shown in figure. A light ray is incident at angle 45º. Find total deviation produced by combination of mirror after two successive reflection. Solution : deviation at mirror deviation at mirror 180º - 2 × 15º = 150º  total deviation 150º - 90º = 60º  Images Formed By Two Mirrors Case 1: When the mirrors are parallel to each other Above figure shows image formed by object placed at distance y, from M1 and at distance x from M2. Number of image formed by parallel plane mirrors is infinity. Case 2: (when the mirrors are inclined at angle ) 1. All the images formed by two mirrors lies on circle have centre C. Here if angle between mirror is  then image will formed on circle at angle (2 - ). If angle  is less number of image formed will be more. 2. If n is number of images, in n then If (i) is even. (ii) is odd and object is kept symmetrically for all other condition SPHERICAL MIRRORS Spherical mirrors are part of polished spherical surface 1. Center of curvature (C) : Center of circle of which mirror is a part 2. Radius of curvature (R) : Radius of circle of which mirror is a part 3. Pole (P) : Centre of mirrors reflecting portion 4. Principal axis : Line which join pole to the centre of curvature 5. Diameter of mirror : Shortest distance between two ends of mirror. Concept of focus : According to figure, Here So, Also, When and and Paraxial Ray : Rays whose angle of incidence are small are known as paraxial rays. Focus : If paraxial rays are parallel to the axis of mirror they will meet or appear to meet at a point on the axis, the point is known as focus and the distance between pole and focus is called is focal length Focal plane : Plane to principal axis and passing through focus is known as focal plane Sign Convention : 1. Pole is taken as origin and principal axis is taken as +x axis 2. Direction of incident light is taken as direction of +ve x axis 3. Object focus, image are referred by their co-ordinate 4. Height above principal axis is taken as positive Mirror Formula 1. Mirror formula holds only for paraxial rays. 2. Proof : (for point object and concave mirror) In …(i) In …(ii) From eqn. (i) and (ii), …(iii) and are very small and Substitute the value in equation (iii), we will get using, and Note : Similarity proof for convex mirror RAY TRACING : 1. (i) Parallel rays passes through focus (ii) Rays through focus goes parallel 2. Ray passing through center of curvature returns in same path. Ex. Ex. 3. In case of convex mirror for real object, image is erect diminished virtual and between pole and focus 4. For concave mirror Object Image Type  C C F F P F C C –  –  P Inverted, real diminished Inverted, real enlarged Erect virtual, large Magnification (m) : Proof : and are similar Illustration : A concave and convex mirror of focal length 10 cm and 15 cm are placed at distance 70 cm. An object AB of height 2 cm is placed at distance 30 cm from concave mirror. First ray is incident on concave mirror then on convex mirror. Find size position and nature of image. Solution : For concave mirror, Using cm Now, cm Image formed by first reflection will be real inverted and diminished For convex mirror Using cm Now, cm Final image will be virtual inverted and diminished Relation between object and image velocity : Differentiation equation velocity of image w.r.t. mirror velocity of object w.r.t. mirror Illustration : A mirror of radius of curvature 20 cm and an object which is placed at distance 15 cm both are moving with velocity 1 ms-1 and 10 ms-1 as shown in figure. Find the velocity of image. Solution : Using Now, using So the image will move with velocity 45 cm/s. Size of image for small size object Differentiating w.r.t. t will get Illustration : An object AB is placed on the axis of concave mirror of focal length 10 cm end A of the object is at 30 cm from mirror. Find the length of the image (I) If length of object is 5 cm (II) If length of object is 1 mm Solution : For point A, Using Similarly for point B using we will get Now size of image (II) here using Now, m So the length of the image is 2.5 × 10-4 m. REFRACTION OF LIGHT 1. Change in the speed of light as it passes from one medium to other is called refraction 2. Light is deviated due to medium when it is not along the normal 3. Velocity of light in medium relative permeability of medium velocity of light in vacuum relative permitivity of medium 4. Absolute refractive index of medium is defined by velocity of light in medium. SNELL’S LAW : (N)  Normal (I)  Incident Ray (R)  refracted Ray 1. V1 and V2 are velocity of light in first and second medium. 1 and 2 are refractive index in first and second medium. 2. For normal incident Snell’s law in general form : 1. For different medium of RI …….. ……. constant constant 2. If refractive index of a medium is function of then normal should be along x axis. 3. If refractive index of a medium is function of then normal should along y axis. Illustration : The ray of light is incident at origin just along y axis shown in the figure the refractive index of the medium varies as . Find the equation of ray inside the medium where r is a constant. Solution : Here constant = k when, Now for point (x, y) here  is from x axis Now, Now solve the above equation to get equation of trajectory. CRITICAL ANGLE : Refractive index of denser medium Refractive index of rarer medium. 1. When a ray propagates from denser medium to rarer medium for 90º angle of refraction corresponding angle of incidence is known as critical angle. 2. At critical angle refraction takes place 3. Critical angle does not depends on angle of incidence. 4.  - i GRAPH angle of deviation, angle of incidence (I) For ray going from denser medium to rarer medium : (II) For light ray going from rarer medium to denser medium : Illustration : A slab of refractive index  is placed in air, light is incident at maximum angle 0 from vertical. Find minimum value of  for which total internal reflection takes place at vertical surface. Solution : For vertical surface  > C …(i) For horizontal surface, …(ii) From eq (i) and (ii) so minimum value of REFRACTION AT SPHERICAL SURFACE : In OCM, i …(i) In CMI, …(ii) According to Snell’s law, For paraxial ray, From eq. (i) and (ii), Note : 1. The formula is applicable only for paraxial rays. 2. When surface is plane, Magnification by curved surface : From Snell’s law, For paraxial rays, So, magnification, Illustration : A linear object of length 4 cm is placed at 30 cm from the plane surface of hemispherical glass of radius 10 cm. The hemispherical glass is surrounded by water. Find the final position and size of the image. Solution : For 1st surface cm, and cm, using cm Using cm. behaves as the object for plane surface and Solving it we will get, cm Now using, cm The final images in all the above cases are shown in figure. PRISM : Prism is combination of two plane refracting surface angle of prism angle of incidence angle of emergence angle of deviation …(i) deviation Using equation (i), …(ii)  - i graph Minimum Deviation : For minimum deviation , So, …(i) Also, …(ii) Using Snell’s law, Maximum Deviation : At maximum deviation and To find imin in different condition and Using Snell’s law, 1. If then 2. If then 3. If then 4. For light ray comes out. 5. For T.I.R. occurs. Thin Prism : For thin prism , and, …(i) deviation using eqn. (i) and Illustration : A prism has refracting angle equal to . It is given that  is the angle of minimum deviation and  is the deviation of the ray entering at grazing incidence. Prove that . Solution : Applying condition of minimum deviation Using , Squaring, …(i) Deviation at grazing incidence, or …(ii) squaring equation (ii) Using …(iii) From equation (i) and (iii) Dispersion of Light : 1. The angular splitting of a ray of white light into a number of components and spreading in different direction is called dispersion of light. 2. Angle of dispersion : Angle between the rays of the extreme colours in the refracted (dispersed) light is called angle of dispersion. 3. Dispersive power () of the medium of the material of prism is given by 4. For small angled prism here and refractive index of material for violet, red and yellow colours respectively. If is not given in the problem then take 5. Cause of dispersion : Dependence of n of  according to cauchy’s formula Combination of two prism : (i) Achromatic Combination : It is used for deviation without dispersion. For this combination Net mean deviation Also and where and are dispressive powers for the two prisms and and are the mean deviation produced by them. (ii) Direct vision combination It is used for producing dispersion without deviation, condition for direct vision combination is Deviation of mean ray Net angle of dispersion or LENS 1. A lens is a homogenous transparent medium (such as glass bounded by two curved surface of one curved and one plane surface. Types of Lens : Assume are Radius of curvature of first and second surface (i) Biconvex Lens : (ii) Plano convex lens (iii) Convexo Concave lens (iv) Biconcave Lens (v) Plane concave lens (vi) Concaveo – convex lens 2. Center most point (P) of lens is known as optical center 3. Line joining both the center of curvature and optical center is called principal axis. 4. (i) Convex Lens Converging (ii) Concave Lens Diverging (I) Real focus : (2nd focus, Main focus) converging point of parallel beam of light is known as real focus (II) 1st Focus : If ray coming from a point or converging to a point after refraction become parallel then the point is known as 1st focus. Note : (1) For convex lens (2) For concave lens Lens Maker’s Formula : refractive index of surrounding. refractive index of lens For surface of radius R1 : …(i) and for surface of radius R2 : …(ii) adding equation (i) and (ii) …(iii) If then …(iv) (Lens Maker’s Formula) From equation (iii) and iv)  Lens Formula Note : (Condition for application of Len’s makers formula) 1. Medium an both side of lens should be same 2. Lens should be thin 3. Light rays should be paraxial Ray Tracing : 1. Ray incident on pole of lens goes undeviated because middle of lens may be treated as parallel slab which cause zero deviation and lateral shift is also negligible for paraxial ray. 2. Ray through optical center passes through without deviation 3. Ray parallel to axis of lens passes through focus. 4. Rays passes through focus goes parallel to principal axis. MAGNIFICATION OF EXTENDED OBJECT : From and So magnification