Consider two coherent sources S1 and S2 producing monochromatic waves to produce an interference pattern. Let the displacement of the wave produced by S1 be given by Y1 = acosωt and the displacement by S2 be Y2 = acos(ωt + φ). Find out the expression for the amplitude of the resultant displacement at a point and show that the intensity at that point will be I = 4a²cos²(φ/2). Hence establish the conditions for constructive and destructive interference. (b) What is the effect on the interference fringes in Young's double slit experiment when (i) the width of the source slit is increased; (ii) the monochromatic source is replaced by a source of white light?

(a)Resultant displacement at the point will be Y = Y1 + Y2 = acos(ωt) + acos(ωt + φ) = 2acos(ωt + φ/2)cos(φ/2)Intensity is the square of the amplitude of displacement I = 4a²cos²(φ/2)For constructive interference, resultant intensity is maximum. Hence, φ/2 = nπ where n is an integer φ = 2nπFor destructive interference, resultant intensity is minimum. Hence, φ/2 = nπ ± π/2 φ = 2nπ ± π(b)(i) When the width of the slit is increased, the intensity of the light waves incident increases and hence, the intensity of the fringes formed increases. However, the fringes formed are not very sharp and increasing the slit width too much spreads the fringes over a greater area. (ii) When a monochromatic light source is replaced by a white light source, fringes formed at a point consist of only a single color (wavelength). Multiple fringes of various colors are formed. The central fringe remains white as all colors constructively interfere there.