Question

Consider two coherent sources s1 and s2 producing monochromatic waves to produce interference pattern read the displacement of the waves produced by sun be given by 1 equal to cos omega t and displacement by is to be y2 equal to

Answer 1

The constructive and destructive interference are given below.

Explanation:

Correct statement:

Consider two coherent sources S1 and S2 producing monochromatic waves to produce interference pattern. Let the displacement of the wave produced by S1 be given by Y 1  = acosωt and the displacement by S2 and 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=4a^2  cos^2  ϕ/2. Hence establish the conditions for constructive and destructive interference.

(a)

Resultant displacement at the point will be

Y = Y 1  +Y  2

​Y = a cos(ωt) + a cos(ωt+ϕ)

Y = 2 a cos (ωt+ϕ/2)cos(ϕ/2)

Intensity is square of amplitude of displacement

I  = 4 a^2  cos^2  (ϕ/2)

For constructive interference, resultant intensity is maximum.

Hence, ϕ/2 = nπ where n is an integer

ϕ = 2 nπ

For destructive interference, resultant intensity is minimum.

Hence, ϕ/2=nπ±π/2

ϕ = 2nπ ± π

Answer 2

Complete question:

Consider two coherent sources S1 and S2 producing monochromatic waves to produce interference pattern. Let the displacement of the wave produced by S1 be given by Y1 = a cos ωt and the displacement by S2 and Y2 = a cos (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 = 4a² cos² Ф/2. Hence establish the conditions for constructive and destructive interference.

Answer:

The condition for constructive interference is 2nπ.

The condition for destructive interference is 2nπ ± π.

Explanation:

The resultant displacement is given at the point:

Y = Y1 + Y2

Where,

Y1 = a cos ωt and Y2 = a cos (ωt + Φ)

Y = (a cos ωt) + (a cos (ωt + Φ))

∴ Y = 2a cos (ωt + Φ/2) cos (Φ/2)

From question, the intensity is given as:

I = 4a² cos² Ф/2

For constructive interference, resultant intensity is maximum.

Ф/2 = nπ

Where, n is integer.

∴ Ф = 2nπ

For destructive interference, resultant intensity is minimum.

Ф/2 = nπ ± π/2

Where, n is integer.

∴ Ф = 2nπ ± π