Wave Optics Class 12 Notes CBSE Physics PDF

Revision Notes Class - 12 Physics Chapter 10 – Wave Optics 1. WAVE FRONT A light source is a point which emits disturbance in all directions. In a homogeneous medium, the disturbance reaches all those particles of the medium in phase, which a...

Revision Notes Class - 12 Physics Chapter 10 – Wave Optics 1. WAVE FRONT A light source is a point which emits disturbance in all directions. In a homogeneous medium, the disturbance reaches all those particles of the medium in phase, which are located at the same distance from the source of light and hence at all the time, every particles must be vibrating in phase with each other. The locus of all the particles of medium, which at any instant are vibrating in the same phase, is called the wave front. Depending upon the shape of the source of light, wave front can be the following types: 1.1 Spherical wave front A point source of light produces spherical wave front. This is because, the locus of every points, which are equidistant from the point source, is a sphere figure (a). 1.2 Cylindrical wave front If the light source is linear (such as a slit), it produces a cylindrical wave front. Here, every points, which are equidistant from the linear source, lie on the surface of a cylinder figure (b). Class XII Physics www.vedantu.com 1 1.3 Plane wave front A wave front will appear plane if small part of a spherical or a cylindrical wave front I originating from a distant source. So it is called a plane wave front figure (c). 1.4 Ray of light The path along which light travels is known as a ray of light. If we draw an arrow normal to the wave front and which points in the direction of propagation of disturbance represents a ray of light. In a ray diagram, thick arrows represent the rays of light. It is also called as the wave normal because the ray of light is normal to the wave front. Key points If we take any two points on a wave front, the phase difference between them will be zero. 2. HUYGENS’S PRINCIPLE Huygens’s principle is a geometrical construction, which can be used to obtain new position of a wave front at a later time from its given position at any instant. Or we can quote that this principle gives a method gives an idea about how light spreads out in the medium. It is developed on the following assumptions: 1. All the points on a given or primary wave front acts as a source of secondary wavelets, which sends out disturbance in all directions in a similar manner as the primary light source. 2. The new position of the wave front at any instant (called secondary wave front) is the envelope of the secondary wavelets at that instant. These two assumptions are known as Huygens principle or Huygens’ construction. Class XII Physics www.vedantu.com 2 Key points Huygens principle is simply a geometrical construction to find the position of wave front at a later time. 3. PRINCIPLE OF SUPER POSITION If two or more than two waves superimpose each other at a common particle of the medium then the resultant displacement ( y ) of the particle is equal to the vector sum of the displacements ( y1 and y 2 ) produced by individual waves.i.e y=y1 +y 2 3.1 Graphical view Class XII Physics www.vedantu.com 3  i. ii. 3.2 Phase/Phase difference/Path difference/Time difference i. Phase: Phase is defined as the argument of sine or cosine in the expression for displacement of a wave. For displacement y = asinωt ; term ωt = phase or instantaneous phase. ii. Phase difference (  ) : Phase difference is the difference between the phases of two waves at a point. i.e. if y1 =a1sinωt and y 2 =a 2sin ( ωt+ ) so phase difference = iii. Path difference () : Path difference between the waves at that point is the λ difference in path length’s of two waves meeting at a point. Also Δ= ×. 2π iv. Time difference (T.D): Time difference between the waves meeting at a point is Class XII Physics www.vedantu.com 4  T given by T.D =  2π 3.3 Resultant amplitude and intensity If we have two waves y1 = a1sinωt and y 2 = a 2sin ( ωt+ ) where a1 ,a 2 = Individual amplitudes,  = Phase difference between the waves at an instant when they are meeting a point. I1 ,I 2 = Intensities of Individual waves. Resultant amplitude: After superimposition of the given waves resultant amplitude (or the amplitude of resultant wave) is given by A= a12 +a 2 2 +2a1a 2cos For the interfering waves y1 =a1sinωt and y 2 =a 2sin ( ωt+ ) , Phase difference between them is 90. So resultant amplitude A= a12 +a 2 2 α ( Amplitude ) 2 Resultant intensity: As we know intensity  I1 -ka12 ,I 2 -ka 2 2 and I=kA 2 (k is a proportionality constant). Hence from the formula of resultant amplitude, we get the following formula of resultant intensity I-I1 +I 2 +2 I1I 2 cos The term 2 I1I 2 cos is called interference term. For incoherent interference this term is zero so resultant intensity I=I1 +I 2. 3.4 Coherent sources Coherent sources are the sources of light which emits continuous light waves with same wavelength, frequency and in phase or having a constant phase difference. 4. INTERFERENCE OF LIGHT If intensity of light at some points is maximum while at some other point intensity is minimum due to the simultaneous superposition of two waves of exactly same frequency (coming from two coherent sources) travels in a medium and in the same direction, this phenomenon is called Interference of light. Class XII Physics www.vedantu.com 5 4.1 Types of Interference Constructive interference Destructive interference Constructive interference is obtained at Destructive interference is obtained at a point when the waves meets at that that point when the wave meets at that point with same phase, ( i.e. maximum point with opposite phase, ( i.e minimum light) light) Phase difference between the waves at  = 180 or ( 2n − 1) ;n = 1,2,... the point of observation  = 0 or 2n. or ( 2n + 1) ;n = 0,1,2... Path difference between the waves at    = ( 2n − 1) ( i.e odd multiple of ) the point of observation Δ=nλ ( i.e. 2 2 λ even multiple of ) 2 Resultant amplitude at the point of Resultant amplitude at the point of observation will be maximum if observation will be minimum a1 =a 2  A min =0 A min =a1 -a 2 a1 =a 2 =a 0  A max =2a 0 If a1 =a 2  A min =0 Resultant intensity at the point of Resultant intensity at the point of observation will be maximum observation will be minimum I max =I1 +I 2 +2 I1I 2 I min =I1 +I2 -2 I1I 2 ( ) ( ) 2 2 I max = I1 + I 2 I min = I1 - I 2 If I1 =I2 =I0  I max =2I0 If I1 =I 2 =I0  I min = 0 4.2 Resultant intensity due to two identical waves 4 The resultant intensity for two coherent sources is given by I=I1 +I 2 +2 I1I 2 cos  For identical source I1 =I 2 =I0 Class XII Physics www.vedantu.com 6    I=I0 +I0 +2 I0 I0 cos  = 4I0 cos 2 2  2  1 + cos  = 2cos 2  Note: ⮚ Redistribution of energy takes place in the form of maxima and minima in interference Imax +I min ⮚ Average intensity: Iav = =I1 +I2 =a12 +a 2 2 2 ⮚ Ratio of maximum and minimum intensities: 2 2 Imax  I1 + I2   I1 / I2 + 1   a1 + a 2   a1 / a 2 + 1  2 2 =    =  =  Imin  I1 − I2   I1 / I 2 − 1   a1 − a 2   a1 / a 2 − 1   I max   +1 I1 a1  I min  Also = = I2 a 2  I max   − 1   I min  ⮚ If two waves having equal intensity ( I1 =I 2 =I0 ) meets at two locations P and Q with path difference 1 and  2 respectively then the ratio of resultant intensity at point 1 cos 2  1  cos 2   I P and Q will be p = 2 =    IQ cos 2 2    cos 2  2  2    5. YOUNG’S DOUBLE SLIT EXPERIMENT (YDSE) An interference pattern is obtained on the screen when monochromatic light (single Class XII Physics www.vedantu.com 7 wavelength) falls on two narrow slits S1 and S2 which are very close together acts as two coherent sources, and when waves coming from these two sources superimposes on each other. Alternate bright and dark bands obtained on the screen in this experiment. These bands are called Fringes. d = Distance between slits. D = Distance between slits and screen = Wavelength of monochromatic light emitted from source. 1) At central position  = 0 or  =0. So, Central fringe will be always bright. 2) The fringe pattern formed by a slit will be brighter than that due to a point. 3) The minima will not be complete dark if the slit widths are unequal. So, uniform illumination occurs for very large width. 4) No interference pattern is observed on the screen if one slit is illuminated with red light and the other is illuminated with blue light. 5) The central fringe will be dark instead of bright if the two coherent sources consist of object and its reflected image. 5.1 Path difference Path difference between the interfering waves meeting at a point P on the screen is yd given by x= =dsin  where x is the position of point P from central maxima. D Class XII Physics www.vedantu.com 8 For maxima at P : x = nλ Where n = 0, 1, 2,... And for minima at P : x = ( 2n − 1)  2 Where n = 0, 1, 2,... Note: If the slits are horizontal path difference is dcos ,so as  increases, x decreases. But if the slits are vertical, the path difference ( x ) is dsin  , so as  increases,  also increases. Class XII Physics www.vedantu.com 9 5.2 More about fringe λD (i) Every fringes will have equal width. Width of one fringe is β = and angular d λ fringe width θ = d (ii) If the YDSE setup is taken in one medium then changes into another, so    3 changes. E.g. in water  w = a  w = a = a w w 4 1 (iii) Fringe width β  i.e if separation between the sources increases,  decreases. d nD (iv) Position of n th bright fringe from central maxima x n = = n;n = 0,1,2,.. d (v) Position of n th dark fringe from central maxima xn = ( 2n − 1) D = ( 2n − 1)  ;n = 1,2,3,.. 2d 2 (vi) In YDSE, if n 1 fringes are visible in a field of view with light of wavelength 1 , while n 2 with light of wavelength  2 in the same field, then n1λ1 =n 2 λ 2 5.3 Shifting of fringe pattern in YDSE The fringe pattern will get shifted if a transparent thin film of mica or glass is placed in the path of one of the waves. If this film is placed in the path of upper wave, the Class XII Physics www.vedantu.com 10 pattern shifts upward and if the film is placed in the path of lower wave, the pattern will shift downward. D β Fringe shift = ( μ-1) t = ( μ-1) t d λ  Additional path difference = (  − 1) t  If the shift is equivalent to n fringes, then n= ( μ-1) t or t= nλ λ ( μ-1)  Fringe shift is independent of the order of fringe (i.e shift of zero order maxima = shift of n th order maxima)  Also, the shift is independent of wavelength. 6. ILLUSTRATIONS OF INTERFERENCE Interference effects are commonly observed in thin films when their thickness is comparable to wavelength of incident light (If it is too thin as compared to wavelength of light it appears dark and if it’s too thick, this will return in uniform illumination of film). Thin layer of oil on water surface and soap bubbles shows various colours in white light due to interference of waves reflected from the two surfaces of the film. Class XII Physics www.vedantu.com 11 6.1 Thin films In case of thin films, interference occur between the waves reflected from its two surfaces and waves refracted through it. Interference in reflected light Interference in refracted light Condition for constructive interference Condition for constructive (maximum intensity) interference (maximum intensity) Class XII Physics www.vedantu.com 12  λ λ Δ=2μtcosr= ( 2n±1) Δ=2μtcosr= ( 2n ) 2 2 For normal incidence r=0 For normal incident λ 2μt=nλ 2μt= ( 2n±1) So 2 Condition for destructive interference Condition for destructive interference (minimum intensity) (minimum intensity) λ λ Δ=2μtcosr= ( 2n ) Δ=2μtcosr= ( 2n±1) 2 2 For normal incidence 2μt=nλ λ 2μt= ( 2n±1) For normal incidence 2 Note: For interference in visible light, the thickness of the film must be in the order of 10,000A ο 7. DOPPLER’S EFFECT IN LIGHT The phenomenon due to relative motion between the source of light and the observer which causes apparent change in frequency (or wavelength) of the light is called Doppler’s effect. According to special theory of relativity v' 1±v/c = v 1-v 2 /c2 If v = actual frequency, v' = apparent frequency, v = speed of source with respect to stationary observer, c = speed of light. Source of light moves towards the Source of light moves away from the stationary observer ( v  c ) stationary observer ( v  c ) Apparent frequency Apparent frequency  v  v v'=v 1+  and v'=v 1-  and  c  c Apparent wavelength Apparent wavelength  v  v λ'=λ 1-  λ'=λ 1+   c  c Class XII Physics www.vedantu.com 13  Doppler’s shift: If apparent wavelength < (ii)Doppler’s shift: If apparent actual wavelength, spectrum of the wavelength > actual wavelength, radiation from the source of light shifts spectrum of the radiation from the source towards the red end of the spectrum. This of light shifts towards the violet end of is called Red shift Doppler’s shift spectrum. This is called Violet shift v v Δλ=λ Doppler’s shift Δλ=λ. c c 8. DIFFRACTION OF LIGHT The phenomenon of light bending around the corners of an obstacle/aperture whose size is comparable to the size of the wavelength of light. 8.1 Types of diffraction The diffraction phenomenon of light is divided into two types Class XII Physics www.vedantu.com 14  Fresnel diffraction Fraunhofer diffraction In Fresnel’s diffraction, either source or In this case both source and screen are screen or both are at finite distance from effectively at infinite distance from the the diffracting device (obstacle or diffracting device. aperture). Common examples: Common examples: Diffraction at single slit, double slit and Diffraction at a straight edge narrow diffraction grating. wire or small opaque disc etc. 8.2 Diffraction of light at a single slit In case of diffraction at a single slit, we get a central bright band with alternate bright (maxima) and dark (minima) bands of decreasing intensity as shown Class XII Physics www.vedantu.com 15  2λD 2λ (i) Width of central maxima βο = and angular width = d d (ii) The path difference between the waves from the two ends of the aperture is given by Δ = nλ ; where n = 1,2,3,...i.e. dsinθ = nλ as the minima occurs at a point on either side of the central maxima. nλ  sinθ = d (iii) The secondary maxima occurs, where the path difference between the waves λ from the two ends of the aperture is given by Δ= ( 2n+1) ; where 2 λ n=1,2,3,...i.e. dsinθ= ( 2n+1)  sinθ = ( 2n+1) λ 2 2d 8.3 Comparison between interference and diffraction Interference Diffraction Produced by the superimposition of Produced by the superposition of waves from two coherent sources. wavelets from different parts of same wave front. (single coherent source) Class XII Physics www.vedantu.com 16 All fringes are of the same width All secondary fringes are of same width but the central maxima has λD double the width β= d λD βο = 2β = 2 d All fringes have equal intensity Intensity decreases as the order of maximum increases. Intensity of all minimum may be zero. Intensity of minima is not zero. Positions of n th maxima and minima. Positions of n th secondary maxima and nλD X n(bright) = λD d X n(Bright) = ( 2n+1) d λD X n(Dark) = ( 2n-1) d nλD X n(Dark) = d Path difference for n th maxima For n th secondary maxima  = n λ Δ= ( 2n+1) 2 Path difference for n th minima Path difference for n th minima  = n  = ( 2n − 1)  8.4 Diffraction and optical instruments Objective lens of instrument like telescope or microscope etc. acts like a circular aperture. By diffraction of light at the circular aperture, a converging lens doesn’t form a point image of an object rather it produces a brighter disc surrounded by alternate dark and bright concentric rings known as Airy disc. Class XII Physics www.vedantu.com 17  1.22λ The angular half width of Airy disc =θ= (where D = aperture of lens) D The lateral width of the image = f (where f = focal length of the lens) Note: Diffraction of light limits the ability of optical instruments to form clear images of objects when they are close to each other. 9. POLARIZATION OF LIGHT Light travel as transverse EM waves. While comparing to magnitude of magnetic field, the magnitude of electric field is much larger. We generally describe light as electric field oscillations. 9.1 Unpolarized light Light with electric field oscillations in every directions in the plane perpendicular to the propagation of it is called Unpolarised light. The oscillation of light is divided into horizontal and vertical component. Class XII Physics www.vedantu.com 18 9.2 Polarized light Polarized or plane polarized light is the light with oscillations only in one plane is. 1. Plane of oscillation is the plane in which oscillation occurs in the polarized light. 2. Plane of polarization is the plane perpendicular to the plane of oscillation. 3. By transmitting through certain crystals such as tourmaline or Polaroid light can be polarized. 9. 3 Polarization by Scattering If a beam of white light is passed through a medium having particles with size comparable to the order of wavelength of light, then the beam will get scattered. This scattered light propagates in a direction perpendicular to the direction of incidence, and it will be plane polarized (as detected by the analyzer). This is called polarization by scattering. Class XII Physics www.vedantu.com 19 9.4 Polarization of Light by Reflection If unpolarized light is reflected using a surface, the reflected light can be obtained as completely polarised, partially polarized or unpolarized. The nature of reflected light depends on the angle of incidence. Polarizing angle or Brewster’s angle ( i p ) is the angle of incidence when the reflected light is completely plane polarized. Class XII Physics www.vedantu.com 20 9.5 Polaroid The device used to produce the plane polarised light is known as a Polaroid. It is based on the principle of selective absorption. Also, it is more effective than the tourmaline crystal. It can also be described as a thin film of ultramicroscopic crystals of quinine idosulphate which has its optic axis parallel to each other. (i) A Polaroid only allows light oscillations which are parallel to the transmission axis to pass through them. (ii) Polarizer is the crystal or Polaroid on which unpolarised light is incident. Crystal or polaroid on which polarised light is incident is called analyzer. Class XII Physics www.vedantu.com 21 Note: If an unpolarized light is passed through a polarizer, the intensity of the transmitted polarized light will become half of the intensity of unpolarised light. (iii) Polaroids are used in making wind shields of automobiles, sun glasses etc. They helps to reduce head light glare of cares and improve colour contrast in old paintings. Polaroids are also used in 3-D motion pictures are in optical stress analysis. 9.6 Malus law The intensity of a polarised light passed through an analyser will change as the square of the cosine of the angle between the plane of transmission of the analyser and the plane of the polariser. This is known as Malus law. Class XII Physics www.vedantu.com 22 I=Iοcos 2θ and A 2 =A ο 2cos 2θ  A=A οcosθ If θ=0ο ,I=Iο ,A=A ο Iο A If θ=45ο ,I= ,A= ο 2 2 If θ=90ο ,I=0,A=0 (i) If Ii = Intensity of unpolarised light. Ii So, I = i.e. if an unpolarised light is converted into plane polarized light ( say 2 by passing it through a Polaroid or a Nicole-prism), its intensity becomes half and I I= i cos 2θ 2 Note: Percentage of polarisation = ( Imax − Imin )  100 ( Imax + Imin ) 9.7 Brewster’s law When a beam of unpolarised light is reflected from a transparent medium (having refractive index =  ), the reflected light will be completely plane polarised at a certain angle of incidence (called the angle of polarisation p ). This is known as Brewster’s law. Also  = tan p --- Brewster’s law i. For iθ p Class XII Physics www.vedantu.com 23 Both reflected and refracted rays becomes partially polarized i. For glass θ p  57ο , for water p  53 10. VALIDITY OF RAY OPTICS By diffraction of light travels, a parallel beam of light travels up to distances as large as few meters can be broadened. 10.1 Fresnel Distance The minimum distance a beam of light can travel before its deviation from straight line path becomes significant/ noticeable is known as Fresnel distance. a2 ZF = λ As wavelength of light is very small, the deviation will be also very small and light can be assumed as travelling in a straight line. So, we can neglect broadening of beam due to diffraction up to distances as large as a few meters, i.e., we can assume that light travels along straight lines and ray optics can be taken as a limiting case of wave optics. Therefore, Ray optics can be considered as a limiting case of wave optics. Class XII Physics www.vedantu.com 24 11. RESOLVING POWER If two point objects are close to each other, images diffraction patterns of those objects will be also close and overlap each other. Limit of resolution of the instrument is the minimum distance between two objects which can be seen separately by the object instrument. 1 Resolving power (R.P) = Limit of Revolution 11.1 Resolving power of Microscope 2 sin  R.P. of microscope =  11.2 Resolving power of Telescope 1 D R.P of telescope = = dθ 1.22λ Where D is aperture of telescope. Class XII Physics www.vedantu.com 25 Class XII Physics www.vedantu.com 26