Derive the expression for the intensity at a point where interference of light occurs. Arrive at the conditions for maximum and zero intensity.

Let the two waves S 1 and S 2 from the two coherent sources interfere at a point P. Let their amplitudes be a and the initial phase difference between them be ϕ . Then the displacements produced by S 1 and S 2 will be given as: y 1 = a cos ω t ....(i) y 2 = a cos ωt + ϕ ....(ii) The intensity produced by each wave, I 0 ∝ a 2 ....(iii) The resultant displacement at point P will be: y = y 1 + y 2 y = a cos ω t + cos ω t + ϕ y = 2 a cos ϕ 2 cos ω t + ϕ 2 ......(iv) From (iii), it is clear that the amplitude of the resultant displacement is 2 a cos ϕ 2 and therefore the intensity at that point will be: I ∝ 4 a 2 cos 2 ϕ 2 On using (iii), we get: I = 4 I 0 cos 2 ϕ 2 .....(v) Where I 0 is the intensity of each of the individual sources. Using (v) , For ϕ = 0 , ± 2 π , ± 4 π . . . . . , the intensity will be maximum and given by: I = 4 I 0 For ϕ = ± π , ± 3 π , ± 5 π . . . . , the intensity will be zero: I = 0