# n identical light waves each of intensity I are superposed in a homogeneous medium. The resultant intensity in their incoherent and coherent addition are respectively (1)zero and n^2 I(2)n I and n^2 I(3)n I and zero(4)n^2 I and n^2 I

Saurabh Koranglekar
4 years ago
Vikas TU
14149 Points
4 years ago
Dear student
If the potential difference between two waves is zero or is constant w.r.t time, then the two ways are said to be coherent.
The waves are non-coherent if the potential difference between the two ways keeps on changing. Lightbulb, study lamp are the examples of the coherent waves. They emit waves at random potential difference.
Now let us consider there are two needles say S1 and S2 moving up and down on the surface of the water and are pointing at point P. So the path difference here is given as  S1P –  S2P. Now the displacement by two needles and S1 S2 are:
y1 = A cos wt ……………… (1)
y2 = A cos wt …………….. (2)
So the resultant displacement at point P is, y =  y1  + y2. When we substitute the value of  y1  and y2  we write,
y = A cos wt + A cos wt
y = 2A cos wt……………….. (3)
Now, we know the intensity is proportional to the square of the amplitude waves.
I0  ∝ A²
Where I0  is the initial intensity and  A² is the amplitude of the wave. From equation 3, we say that A = 2A. So,
I0  ∝ (2A)² or I0  ∝ 4 A²
I = 4 I0
Now, if two needles that are S1 are S2 are in the same phase, the potential difference is,
S1P  – S2P = nλ
Where n = 0, 1, 2,3 ……… and λ = the wavelength of the wave. If the two needles  S1 and S2 are vibrating at its destructive interference then, the potential difference is
S1P –  S2P = (n + 1/2) λ
Now if the potential difference of the waves is Φ then,
y1  = α cos wt
y2 = α cos wt
The individual intensity of each wave is  I0 , we get,
y =  y1  + y2
= α cos wt + α cos (wt +Φ)
y = 2 α cos(Φ/2) cos (wt + Φ/2)
Since, the intensity is I0 ∝ A²
I0 ∝ 4α² cos² (Φ/2)
I = 4 I0  cos² (Φ/2)
Well, the time-averaged value of cos²(Φt/2) is 1/2. So, the resultant intensity will be  I = 2 I0 at all the points.