Problem 34 - Master Resource Book in JEE Main Physics

Question

In Young's double slit experiment, the two slits act as coherent sources of equal amplitude \(A\) and wavelength \(\lambda\). In another experiment with the same setup, the two slits are sources of equal amplitude \(A\) and wavelength \(\lambda\) but are incoherent. The ratio of the intensity of light at the mid- point of the screen in the first case to that in the second case is (a) \(2: 1\) (b) \(1: 2\) (c) \(3: 4\) (d) \(4: 3\)

Step-by-Step Solution

Answer

The ratio of intensities is \(2:1\).

Step 1: Understanding Coherent Sources

In the first experiment, the two slits are coherent sources. This means the waves from the slits maintain a constant phase relationship. At the midpoint of the screen, the waves interfere constructively, leading to maximum intensity. The resultant amplitude at this point is given by the formula \[\text{Resultant Amplitude} = 2A\]where \(A\) is the amplitude from each slit. Thus, the intensity (\(I\)) at the midpoint is calculated as follows:\[I_1 = \left(2A\right)^2 = 4A^2.\]

Step 2: Understanding Incoherent Sources

In the second experiment, the slits act as incoherent sources. This means the waves do not maintain a constant phase relationship, leading to random interference and the overall intensity is simply the sum of individual intensities of the waves. The individual intensity from each slit is \[I = A^2.\]Thus, the total intensity at the midpoint is the sum of the individual intensities from each slit:\[I_2 = A^2 + A^2 = 2A^2.\]

Step 3: Calculating the Intensity Ratio

Having obtained the individual intensities for coherent and incoherent cases, we calculate the ratio of the intensity of light at the midpoint of the screen for both cases:\[\text{Intensity Ratio} = \frac{I_1}{I_2} = \frac{4A^2}{2A^2} = \frac{4}{2} = 2:1.\]

Key Concepts

Coherent Sources

In the context of Young's double slit experiment, coherent sources refer to two light sources that emit waves with a constant phase difference. In simpler terms, this means that the peaks and troughs of the waves from these sources line up perfectly over time. This synchronization is crucial for producing predictable interference patterns, such as those observed on the screen in Young's experiment.
The key characteristics of coherent sources include:

• Consistent phase relationship: The waves maintain a stable phase difference, allowing for constructive and destructive interference.
• Same frequency and wavelength: Coherent sources emit light waves of the same frequency and hence, of the same wavelength.

In Young's double slit experiment, if the slits act as coherent sources, the interference at the midpoint results in maximum light intensity. The waves superimpose to double the amplitude, making the light appear brighter at certain points. This phenomenon is what causes the distinctive bright and dark bands in interference patterns.

Incoherent Sources

Unlike coherent sources, incoherent sources emit waves that do not maintain a fixed phase relationship. This means that the peaks and troughs of waves from these sources do not overlap consistently over time. As a result, the interference pattern becomes unpredictable and leads to a more uniform distribution of light intensity across the field.
The main features of incoherent sources are:

• Variable phase relationships: The phase difference between the waves changes randomly, leading to inconsistent interference.
• Random interference: Since the waves do not synchronize, any superimposition results in a less pronounced or uniform pattern.

In Young's experiment, when the slits act as incoherent sources, the light intensity at the midpoint is merely the sum of the intensities from each slit. This is because the random phase differences prevent the waves from effectively reinforcing each other to create the bright fringe typically noted in coherent lighting scenarios.

Light Intensity

Light intensity in the context of wave interference refers to the power per unit area received from a wave. It is proportional to the square of the wave's amplitude, making intensity an important measure of brightness in interference patterns.
Factors influencing light intensity in the setup of Young’s double slit experiment include:

• Amplitude: Greater amplitude results in higher intensity. When two coherent waves interfere constructively, their amplitudes add up leading to a brighter light at certain points.
• Interference Type: Constructive interference increases intensity, whereas destructive interference decreases it, as waves cancel each other out.
• Type of sources: Coherent sources create higher intensity peaks due to effective wave overlapping. In contrast, incoherent sources result in lower intensity when compared to their coherent counterparts.

In Young's experiment, the coherent case doubles the amplitude leading to an intensity four times greater than in the incoherent case, explaining the intensity ratio of 2:1 at the midpoint of the screen.