Title 19. Ray Optics and Wave Optics.pdf ✅

Ray Optics and Wave Optics 417 19 Ray Optics and Wave Optics QUICK LOOK Mirrors are objects that reflect light in a way that is most proportionate to the original object. Certain mirrors also aim to filter out wavelengths; this projects a different image as shown on the mirror. The most commonly found mirrors are plane mirrors which have a flat surface, which are found in most households. There are also curved mirrors which distort the reflected image. In most vehicles mirrors is a necessity as they can help locate blind spots, allow for drivers to check before merging lanes, help Drivers Park properly and many more. On larger vehicles such as trucks and buses, curved mirrors are especially useful for helping drivers in minimizing blind spots. There would be a lot more accidents if mirrors were not a part of vehicles. Dentists also make good use of mirrors by finding those nasty decays located in the inner depths of one’s mouth. Those mirrors can either be curved or flat. Besides dentists, mechanics are also able use such mirrors to view parts of the machinery which are very inaccessible. Reflection Law: Angle of incidence (i) = angle of reflection ( ) r′ Deviation produced in reflection, δ = ° − (180 2i) Plane Mirror The focal length of a plane mirror is infinity. The image formed by plane mirror is virtual, erect, of the same size as object, but laterally inverted. Figure: 19.1 For incident ray fixed, if plane mirror is rotated through an angle θ , he reflected ray rotates through an angle 2 . θ If object approaches mirror with speed v, the relative velocity of approach of image towards object is 2v. To see full image of him-self in a plane mirror, its length of mirror is just half of height of the man. A thick mirror forms a number of images, out of which the second image is brightest. The minimum size of plane mirror to be fixed on the wall of a room, so that an observer in the middle of the room can see the full image of the wall behind him is one-third of the height of wall. Number of images formed by two inclined mirrors 360 N θ ° = if 360 θ ° is odd. 360 1 θ ° = − if 360 θ   °     is even end object lies symmetrically between mirrors. Deviation produced by tow plane mirrors inclined at an angle θ δ θ , 360 2 = ° − (independent of angle of incidence). Spherical Mirrors If r is radius of curvature and f is focal length, then r f = 2 . Mirror formula 1 1 1 f u ν = + Lateral magnification, 0 I f m u u f ν = = = − − Axial magnification 2 2 2 0 I f m u u f ν   = = − = −    − Area magnification 2 1 2 0 A m A u ν = = − Newton’s formula for real image 2 1 2 x x f = , where 1 x = distance of object form focus, 2 x = distance of image form focus Magnification due to a Mirror Transverse or linear magnification m is defined as ( ) ( ) size of image size of image l m O = It can be proved that for both the concave and convex mirrors: m u ν = − or f m f u = − − or f f −ν If m >1, the image is enlarged, If m =1, the image is of same size. p p′ Object Virtual image
418 Quick Revision NCERT-PHYSICS If m <1, the image is reduced, If m > 0, the image is erect, If m < 0, the image is inverted. If object lies along the principal axis, then its magnification is called the longitudinal magnification.It is given by 2 m 2 u ν = − Table: 19.1 Image Formation form a Concave Mirror Position of Object Position of Image Nature of Image At infinity At focus (in front of mirror) Size of Image: Point size (smaller than object) Real and inverted Between infinity and centre of curvature C Between centre of curvature C and focus F (in front of mirror) Size of Image: Smaller than that of object Real and inverted At the centre of curvature At the centre of curvature (in front of mirror) Size of Image: Of the same size as that of the object Real and inverted Between the centre of curvature C and focus F Between infinity and the centre of curvature C (in front of mirror) Size of Image: Bigger than the size of object Real and inverted At the focus F At infinity (in front of mirror) Size of Image: Bigger that the size of the object Real and inverted Between focus F and pole P Behind the mirror Size of Image: Bigger than the object Virtual and erect Note For a concave mirror, if object lies between pole and focus, then only image is virtual, erect and enlarged. Image Formation from a Convex Mirror: For a convex mirror, m is +ve and less than one, i.e., the image is virtual, erect and diminished in size, for all positions of the object. Refraction: In refractions when light ray enters form one medium to another, its frequency remains unchanged. Snell’s law 2 1 1 1 1 1 2 2 sin sin i v r v μ λ μ μ λ = = = = Deviation caused by refraction δ = − (i r) 2 1 1 2 1 μ μ = 1 3 2 3 1 2 μ μ μ = Lateral Shift Lateral shift sin( ) cos t i r x r − = real depth apparent depth t t x μ = = − ∴ apparent shift 1 x t 1 μ   = −     Refractive index ari medium c v λ μ λ = = Refraction at spherical surface μ μ μ μ 2 1 2 1 − = = v u R Formula for refraction through a thin lens ( ) 1 2 1 2 1 1 1 μ 1   = − −     f R R (Proper signs of R1 and R2 are to be used) Figure: 19.3 Air Water Point Real depth Apparent depth Image of point F C Figure: 19.2 F P C F P F P C F P C F P C F P C
Ray Optics and Wave Optics 419 Special Cases For a double convex lens, ( ) 1 2 1 2 1 1 1 1 f R R μ   = − +     For a double concave lens ( ) 1 2 1 2 1 1 1 1 f R R μ   = − − +     (signs of R1 and R2 have been used) If media on two sides of lens are different then, 3 3 2 2 1 1 2 f R R μ μ μ μ μ − − = + Formation of image by a lens 1 1 1 f v u = − Lateral magnification, I v f m O u u f = = = + Axial magnification 2 2 x 2 v f m u u f   = =     + Power of lens ( ) 1 in meter P f = Diopter Equivalent Focal Length Lenses in contact 1 2 1 1 1 F f f = + 1 2 1 2 f f F f f = + or P P P = +1 2 Lenses at separation: 1 2 1 2 1 1 1 1 F f f f f = + − 1 2 1 2 f f F f f d = + − or P P P dPP = + = 1 2 1 2 Newton’s formula: If 1 x and 2 x are distances of object and image form first and second principal focus, then 1 2 1 2 f f x x = . For same medium on both side 2 1 2 x x f = − Displacement Method: Size of object O I I = 1 2 and 2 2 4 D x f D − = and 2 1 2 2 I v I u = Lens immersed in a liquid: denotes glass 1 denotes liquid 1 denotes air a g liquid air a g a g g f f l a μ μ μ   = −  = × =  −  =  Special Cases If 1 , a g a μ μ > the nature of lens remains unchanged; but focal length changes. If 1 , a g a μ μ < the nature of lens changes. If 1 1 , a g a μ μ = → ∞ f and lens behaves as a plate. Silvering at a Lens Surface: If a convex lens is silvered at back surface, it behaves as a concave mirror of equivalent focal length 1 2 2 F f f l m = + . Where l f is focal length of un silvered lens and m f is focal length of silvered face acting as mirror, 2 2 m R f = . Special Cases Plano-convex lens silvered at plane surface m f = ∞ 1 2 1 ( ) F R μ − = ⇒ 2 1 ( ) R F μ = − Plano-convex lens silvered at convex surface 1 2 f R μ = ⇒ 2 R F μ = A concave lens placed in contact with a plane mirror behaves as a convex mirror of focal length 2 f . Formation of Image by Convex Lens Figure: 19.4 2F1 F1 0 F2 2F2 2F1 F1 0 F2 2F2 2F1 F1 0 F2 2F2 Object beyond 2F Image between F and 2F Object at infinity Image at F Object at 2F Image at 2F 2F1 F1 0 F2 2F2 Object at F Image at infinity 2F1 F1 0 F2 2F2 Object between F and 2F Image on the same side behind object Object between F and 2F Image beyond 2F 2F1 F1 0 F2 2F2 O i f f h' h Object size Image size P = 1/f (f in meters) Common Gaussian form of lens equation: 1 1 1 O i f + = Linear magnification: i h M O h − ′ = =
420 Quick Revision NCERT-PHYSICS Table 19.2: Position of Object and Image for Convex Lens Position of object Position of image Size of image Nature of image At infinity At focus Point size Real and inverted Beyond 2F Between F and 2F Smaller in size Real and inverted At 2F At 2F Same size Real and inverted Between F and 2F Beyond 2F Bigger in size Real and inverted At F At infinity Bigger in size Real and inverted Between optical centre C and F On the same side as is the object Bigger in size Virtual and erect Formation of Image by Concave Lens Figure: 19.5 Table: 19.3: Position of Object and Image for Concave Lens Position of object Position of image Size of image Nature of image Anywhere One the same side between optical centre C and F Smaller in size Virtual and erect Ray Optics or Geometrical Optics Snell’s law of refraction 1 1 2 2 n n sin sin θ θ = Critical angle of total reflectionθ1c , 2 1 1 sin c n n θ = Figure: 19.6 Minimum Deviation in Prism Figure: 19.7 θ θ θ i i 1 2 1 = = 2 m i δ θ α = − sin sin 2 2 m n α δ α + = Dispersion of prism 2 1 sin cos cos i r n D δ α λ λ θ θ ∂ ∂ = = ∂ ∂ In particular, at minimum deviation λ ∂ = ∂ n D ( ) 2sin 2 1 cos 2 m n α λ δ α = = + Resolving Power: The term resolving power is applied to spectrographic devices using a prism or a grating. Resolving power signifies the ability of the instrument to form separate spectral images of two neighbouring wavelengths, λ and λ λ + d in the wavelength region λ. Resolving power of prism n R λ λ λ ∂ = = − ∆ ∂ l ∆ = λ least difference in wavelength for two lines to be resolved. Examples: Figure: 19.8 Examples: 2 2 2 2 2 cos sin( ) sin r i r d a n n θ θ θ α = − − − (This is a way to measure n1 or n1. θ is slightly less than 90° and n1 < n2) a n1 n2 d n1 θi θr n1 n2 ∝ Slit source Fabry-perot Etalon Prism R G B Blue Green Red Fast medium (smaller index of refraction) Show medium θ1 θ2 n1 n2 Linear magnification: 0 i h M h ′ = = Image size: h′ Object size: h Object inside focal length Common Gaussian form of lens equation: 1 1 1 0 i f + = O f i Object outside focal length