Title 10. WAVE OPTICS.pdf ✅

Newton’s Corpuscular Theory of Light This theory was given by Newton Characteristics of the theory (i) Extremely minute, very light and elastic particles are being constantly emitted by all luminous bodies (light sources} in all directions which are known as corpuscles. (ii) These corpuscles travel with the speed of light. (iii) When these corpuscles strike the retina of our eye then they produce the sensation of vision. (iv) The velocity of these corpuscles in vacuum is 3 × 108 m/s. (v) The different colours of light are due to different size of these corpuscles. (vi) The rest mass of these corpuscles is zero. (vii) The velocity of these corpuscles in an isotropic medium is same in all directions but it changes with the change of medium. (viii) These corpuscles travel in straight lines. (ix) These corpuscles are invisible. The phenomena explained by this theory (i) Reflection and refraction of light. (ii) Rectilinear propagation of light. (iii) Existence of energy in light. The phenomena not explained by this theory (i) Interference, diffraction, polarization, double refraction and total internal reflection. (ii) Velocity of light being greater in rarer medium than that in a denser medium. (iii) Photoelectric effect and Compton effect. Huygens’s Wave Theory of Light This theory was enunciated by Huygens in a hypothetical medium known as luminiferrous ether. Ether is that imaginary medium ·which prevail~ in all space and is isotropic, perfectly elastic and massless. The velocity of light in a medium is constant but changes with change of medium. This theory is valid for all types of waves. (i) The locus of all ether particles vibrating in same phase is known as wavefront. (ii) Light travels in the medium in the form of wavefront. (iii) When light travels in a medium then the particles of medium start vibrating and consequently a disturbance is created in the medium. · (iv) Every point on the wavefront becomes the source of secondary wavelets. It emits secondary wavelets in all directions which travel with the speed of light. (v) The tangent plane to these secondary wavelets represents the new position of wave front. CHAPTER – 10 WAVE OPTICS WAVE OPTICS
The phenomena explained by this theory (i) Reflection, refraction, interference, diffraction. (ii) Rectilinear propagation of light (iii) Velocity of light in rarer medium being greater than that in denser medium. Phenomena not explained by this theory (i) Photoelectric effect and Raman effect. Wavefront The locus of all the particles vibrating in the same phase in known as wavefront. Types of wavefronts The shape of wavefront depends upon the shape of the light source from, the wavefront originates. On this basis there are three types of wavefronts. (i) Spherical Wavefront: If the waves in a medium are originating from a point source, then they propagate in all directions. If we draw a spherical surface centered at point-source, then all the particles of the medium lying on that spherical surface will be in the same phase, because the disturbance starting from the source will reach all these points simultaneously. Hence in this case, the wavefront will be spherical and the rays will be the radial lines. (ii) Cylindrical Wavefront: If the waves in a medium are originating from a line source, then they too propagate in all directions. In this case the locus of particles vibrating in the same phase will be a cylindrical surface. Hence in this case the wavefront will be cylindrical. (iii) Plane Wavefront: At large distance from the source, the radii of spherical or cylindrical wavefront will be too large and a small part of the wavefront will appear to be plane. At infinite distance from the source, the wavefronts are always plane and the rays are parallel straight lines. y = a sin 2π ( t T − x λ ) Characteristic of wavefront The phase difference between various particles on the wavefront is zero. These wavefronts travel with the speed of light in all directions in an isotropic medium. A point source of light always gives rise to a spherical wavefront in an isotropic medium. In anisotropic medium it travels with different velocities in different directions. Normal to the wavefront represents a ray of light. It always travels in the forward direction in the medium. Coherent And Incoherent Sources of Light The sources of light emitting waves of same frequency having zero or constant initial phase difference are called coherent sources. The sources of light emitting waves with a random phase difference are called incoherent sources. For interference phenomenon, the sources must be coherent. Methods of Producing Coherent Sources: Two independent sources can never be coherent sources. There are two broad ways of producing coherent sources for the same source. (i) By division of wavefront: In this method the wavefront (which is the locus of points of same phase) is divided into two parts. The examples are young’s double slit and Fresnel’s biprism. (ii) By division of amplitude: In this method the amplitude of a wave is divided into two parts by successive reflections, e.g., Lloyd’s single mirror method. Interference Of Light When two light waves having same frequency and equal or nearly equal amplitude are moving in the same direction superimpose then different points have different light intensities. At some point the intensity of light is maximum and at some paint it is minimum this phenomenon is known as interference of light. Phasor diagram By right angle triangle: A 2 = (a1 + a2 cos ) 2 + (a2 sin ) 2 Resultant amplitude 2 2 A a a 2a a cos 1 2 1 2  = + +  Phase angle  = 1 2 1 2 a sin tan a a cos −        +  Intensity  (Amplitude)2   A 2  I = KA2 So, 2 1 1 I Ka = and 2 2 2 I Ka =  1 2 1 2 I I I 2 I I cos = + +  Here, 1 2 2 I I cos is known as interference factor If the distance of a source from two points A and B is x1 and x2 then Phase difference 2 1 2 (x x )   = −   2  =  
Time difference t t 2   =  Phase difference 2π = Path difference λ = Time difference T  φ 2π = δ λ = Δt T Types of interference Constructive Interference When both waves are in same phase then phase difference is an even multiple of    = 2n; n = 0, 1, 2, ......... Path difference is an even multiple of 2  2   =    2nπ 2π = δ λ  2n ( λ 2 )   = n (where m= 0, 1, 2,..... When time difference is an even multiple of T 2 ∴ Δt = 2n ( T 2 ) In this condition the resultant amplitude and intensity will be maximum. Amax = (a1 + a2)  Imax = I1 + I2 + 2√I1√I2 = (√I1 + √I2) 2 Destructive Interference When both the waves are in opposite phase.  = (2n – 1) ; n = 1, 2, ..... When path difference is an odd multiple ofλ 2 , δ = (2 n−1) λ 2 , n = 1,2, . . . . . ... Let two waves having amplitude a1 and a2 and same frequency, and constant phase difference  superpose. Let their displacement are: y1 = a1 sin t and y2 = a2sin(t + ). y = y1 + y2 = A sin(t + ). Where A = Amplitude of resultant wave  = New initial phase angle When time difference is an odd multiple of T 2 , Δt = (2n − 1) T 2 , (n =1,2, . . . . . . ) In this condition the resultant amplitude and intensity of wave be minimum A a1−a2)min  I√I1√I2 2 min Young's Double Slit Experiment According to Huygens, light is a wave. It is proved experimentally by YDSE. S is a narrow slit illuminated by a monochromatic source of light sends wave fronts in all directions. Slits S1 and S2 become the source of secondary wavelets which are in phase and of same frequency. These waves are superimposed on each other which give rise to interference. Alternate dark and bright bands are obtained on a screen (called interference fringes) placed certain distance from the plane of slit S1 and S2. Central fringe is always bright (due to path length S1O and S2O to center of screen are equal) and is called central maxima. In YDSE division of wavefront takes place. If one of the two slit is closed, the interference pattern disappears. It shows that two coherent sources are required to produce interference pattern. If white light is used as parent source, then the fringes will be colored and of unequal width. (i) Central fringe will be white (ii) The fringe closest on either side of the central white fringe is red and the farthest will appear blue. After a few fringes, no clear fringe pattern is seen. Condition For Bright and Dark Fringes Bright Fringe D = distance between slit and screen, d = distance between slit S1 and S2 Bright fringe occurs due to constructive interference. For constructive interference path difference should be even multiple of 2   Path difference  = PS2 – PS1 = S2L = (2n) 2  Q. Two sources of intensity I and 3I are used in an interference experiment. Find the intensity at a point where the waves from the two sources superimpose with a phase difference (1) Zero (2) π/2 Sol. We know, I ′ = I1 + I2 + 2√I1 I2 〈cos(δ1 − δ2 )〉 I ′ = I1 + I2 + 2√I1 I2 〈cos(δ)〉 (i) As δ = 0, cosδ = 1 ∴ I ′ = 3I + I + 2√3I × I × 1 I ′ = 4I + 2√3I (ii) As δ = π/2, cosδ = 0 I ′ = 3I + I + 2√3I × I × 0 I ′ = 4I