(a) (i) 'Two independent monochromatic sources of light cannot produce a sustained interference pattern'. Give reason.

(ii) Light waves each of amplitude "a" and frequency "ω", emanating from two coherent light sources superpose at a point. If the displacements due to these waves is given by y₁ = a cos ωt and y₂ = a cos(ωt + ϕ) where ϕ is the phase difference between the two, obtain the expression for the resultant intensity at the point.

(b) In Young's double slit experiment, using monochromatic light of wavelength λ, the intensity of light at a point on the screen where path difference is λ, is K units. Find out the intensity of light at a point where path difference is λ/3.

(a)
(i) The condition for the sustained interference is that both the sources must be coherent (i.e. they must have the same wavelength and the same frequency, and they must have the same phase or constant phase difference).
Two sources are monochromatic if they have the same frequency and wavelength. Since they are independent, i.e. they have different phases with irregular difference, they are not coherent sources.
ii) 

Let the displacement of the waves from the sources S₁ and S₂ at point P on the screen at any time t be given by: 

y₁ = a cos ωt

y₂ = a cos (ωt + Φ)

where, Φ is the constant phase difference between the two waves.
By the superposition principle, the resultant displacement at point P is given by:

y = y₁ + y₂

y = a cos ωt + a cos (ωt + Φ)

  =2 a[cos cos  
y = 2 acos cos               ... (i) 
                         ... (2)

Then, equation (i) becomes:
y = A cos (ωt+

Now, we have:

                     ... (3)

The intensity of light is directly proportional to the square of the amplitude of the wave. The intensity of light at point P on the screen is given by:
I = 4 a² cos² ()                                  ... (4) 

(b) Wavelength of monochromatic light = 

Path difference = .
So, phase difference, 

Intensity of light = K units
Intensity is given by, I = 
When path difference is 
Intensity of light, I’= =