Question

What are coherent sources of light? Draw the variation of- intensity with position, in the interference pattern of Young's double slit experiment.

Or

Describe the condition for constructive and destructive interference.

Answer 1

Coherent sources: Coherent sources are those sources which emit continuously light of the same wavelength and magnitude either in the same phase or with a constant phase difference.

Expression

Let the waves of two coherent sources be

y₁ = a sin ωt

and y₂ = b sin (ωt + Φ), 

where a and b are the respective amplitudes of the two waves and Φ is the constant phase angle by which the second wave leads the first wave [Fig].

According to superposition principle, the displacement y of resultant wave is

y = y₁ + y₂ = a sin ωt + b sin (ωt + Φ)

or y = a sin ωt + b sin ωt cos Φ + b cos ωt sin Φ

or y = sin ωt (a + b cos Φ) + cos ωt. b sin Φ

Substituting a + b cos Φ = A cos θ ..........(1)

and b sin Φ = A sin θ ..........(2)

we have

y = sin ωt. A cos θ + cos ωt. A sin θ

or y = A [sin ωt. cos θ + cos ωt. sin θ]

or y = A sin (ωt + θ),

where A is the amplitude of the resultant wave. Squaring and adding (1) and (2), we have

a² + b² cos² Φ + 2ab cos Φ + b² sin² Φ 

= A² cos² θ + A² sin² θ

or a² + b² (cos² Φ + sin² Φ + 2ab cos θ)

= A² (cos² θ + sin² θ)

or A² = a² + b² + 2ab cos Φ

Since resultant intensity is proportional to square of amplitude,

∴ I ∝ A²

or I ∝ (a² + b² + 2ab cos Φ)

or I = a² + b² + 2 ab cos Φ ............(3)

(For convenience, we assume that intensity of light is equal to square of amplitude)

(i) For constructive interference (or for maximum).

I should be maximum

i.e. cos Φ = 1 and I_max = (a + b)²

or Φ = 0, 2π, 4π, ...........

or Φ = 2π n (where n = 0, 1, 2, ..........)

or Path diff. = \(\frac{\lambda}{2\pi} \phi = \frac{\lambda}{2\pi} \)  2πn = n λ ..........(4)

So, for constructive interference (or for maximum intensity), path difference should be integral multiple of λ.

(ii) For destructive interference (or for minima)

I should be minimum

i.e. cos Φ = -1 and I_min = (a - b)²

or Φ = π, 3π, 5π, ..........................

or Φ = (2n + 1) π (where n = 0, 1, 2, ..........)

or path difference = \(\frac{\lambda}{2\pi} (2n + 1) \pi\)

= (2n + 1) \(\frac{\lambda}{2}\) ...............(5)

So, for destructive interference (or for minimum intensity), path difference should be an odd multiple of λ/2.

Variation of intensity with position

From above discussion, we find that intensity of light is maximum when path difference is nλ i.e. ±λ, 2λ, 3λ .......... and minimum when path difference is (2 n+ 1) λ/2

i.e. ± \(\frac{\lambda}{2}, 3\frac{\lambda}{2}, 5\frac{\lambda}{2}\)

The variation in intensity of light with position in interference is as shown in Fig. (b).