Optics-Questions on Interference

The following MCQ appeared in Kerala Engineering Entrance 2005 test paper:

In the Young’s double slit experiment, the intensity of the central maximum is observed to be I₀. If one of the slits is covered, the intensity at the central maximum will become

(a) I₀/2 (b) I₀/√2 (c) I₀/4 (d) I₀ (e) I₀²

If the resultant amplitude (due to the two interfering waves) at the central maximum is ‘a’. we can write

I₀ α a², since the intensity is proportional to the square of the amplitude.

When one of the slits is covered, the amplitude is reduced to a/2. If ‘I’ is the intensity at the position of the central maximum now, we can write

I α (a/2)².

From the above, we obtain I = I₀/4 [Option (c)].

Now, consider the following question:

In a double slit interference pattern, the intensity at the centre of a bright fringe is I. The intensity at a point one quarter of the distance to the next bright fringe is

(a) I/2 (b) I/4 (c) I/8 (d) I (a) zero

At the centre of a bright fringe the waves arrive in phase. You may imagine that the photons starting from the two slits are in the same state of vibration when they reach the position of the centre of a bright fringe and that is why their amplitudes get added to produce maximum intensity. At the centre of the next bright fringe, the photons will have an extra phase difference of 2π, but this too is ‘in phase’ condition (for the same state of vibration).

At a point one quarter of the distance to the next bright fringe, the phase difference between the interfering photons will be 2π/4 = π/2.

If ‘a’ is the amplitude of each interfering wave, the resultant amplitude at the centre of a bright fringe is 2a and the intensity I is given by

I α 4a²

At a point one quarter of the distance to the next bright fringe, the amplitudes are added with a phase difference of π/2 and the resultant amplitude is √(a² + a²) = √2 a. The intensity (I') in this case is given by

I' α 2a²

From the above expressions, we obtain I' = I/2 [Option (a)].

[Note that the intensity (I) produced by two interfering waves of the same amplitude ‘a’ is given by I α 4a²cos²(δ/2) where ‘δ’ is the phase difference].

You will find more multiple choice questions with solution in this section at physicsplus: Multiple Choice Questions on Wave Optics and at physicsplus: Questions on Polarisation