Problem 79 - Master Resource Book in JEE Main Physics

Question

Assertion In Young's experiment, for two coherent sources, the resultant intensity given by \(I=4 I_{0} \cos ^{2} \frac{\phi}{2}\) Reason Ratio of maximum and minimum intensity \(\frac{I_{\mathrm{max}}}{I_{\min }}=\frac{\left(\sqrt{I_{1}}+\sqrt{I_{2}}\right)^{2}}{\left(\sqrt{I_{1}}-\sqrt{I_{2}}\right)^{2}}\)

Step-by-Step Solution

Answer

The assertion and reason are both correct but the reason does not explain the assertion.

Step 1: Understand the Assertion

The assertion states that in Young's double-slit experiment, the intensity pattern is given by the formula \(I = 4 I_{0} \cos^2 \frac{\phi}{2}\), where \(I\) is the resultant intensity, \(I_{0}\) is the initial intensity of each source, and \(\phi\) is the phase difference between the waves. This formula describes the interference pattern seen in the experiment, indicating that the resultant intensity varies with the cosine square of half the phase difference. This can be rewritten considering trigonometric identities as \(I = 2I_0(1 + \cos\phi) \).

Step 2: Analyze the Reason

The reason provided is a formula for the ratio of maximum and minimum intensities in terms of the amplitudes of two coherent sources: \( \frac{I_{\mathrm{max}}}{I_{\mathrm{min}}} = \frac{(\sqrt{I_{1}}+\sqrt{I_{2}})^{2}}{(\sqrt{I_{1}}-\sqrt{I_{2}})^{2}} \). This formula is derived from the intensities \(I_1\) and \(I_2\) of the two sources, showing how their constructive and destructive interference leads to the max/min intensities.

Step 3: Check Consistency of Assertion and Reason

The assertion and reason need to be checked for logical consistency. While the assertion provides a specific formula for resultant intensity in terms of cosine, the reason addresses the ratio of max and min intensity, using principles from interference. Both discuss interference, but they do not necessarily directly pertain to the same equation or scenario.

Step 4: Conclusion

The assertion makes a specific statement about intensity in Young's experiment while the reason explains another, broader principle of intensity ratios in interference. They are related through the phenomenon of interference but address separate aspects. Thus, while each is true, the reason does not directly explain the assertion.

Key Concepts

Interference Pattern

Young's Double-Slit Experiment involves two coherent light sources creating an interference pattern. This pattern arises when the peaks and troughs of light waves overlap. The light waves either strengthen each other, known as constructive interference, or cancel each other out, known as destructive interference. As a result, a series of bright and dark fringes appear on a screen.

• Constructive Interference: Occurs when the waves are in phase, leading to maximum intensity or bright fringes.
• Destructive Interference: Happens when the waves are out of phase, resulting in minimum intensity or dark fringes.

Each fringe's position is determined by the phase difference of the waves. Thus, understanding the principles of interference is crucial for grasping how the intensity patterns emerge in such experiments.

Resultant Intensity

The resultant intensity in Young's experiment is defined by the equation \(I = 4 I_{0} \cos^2 \frac{\phi}{2}\). Here, \(I\) represents the observable intensity on the screen, \(I_0\) indicates the initial intensities of each source, and \(\phi\) denotes phase difference. This formula reveals the connection between phase difference and intensity: the resultant intensity is directly proportional to the square of the cosine of half the phase difference.

The maximum resultant intensity occurs when the phase difference \(\phi\) is an even multiple of \(\pi\) (constructive interference), whereas minimum intensity is seen when \(\phi\) is an odd multiple of \(\pi\) (destructive interference). Therefore, understanding the resultant intensity allows us to predict where bright and dark fringes will fall on the screen.

Phase Difference

Phase difference is a vital concept in determining the fringe patterns in Young's Double-Slit Experiment. It describes the offset between the peaks of two wave fronts and is symbolized by \(\phi\). The phase difference can result in different levels of interference.

• Zero Phase Difference: Identical wave peaks result in constructive interference and maximum intensity.
• Phase Difference of \(\pi\): Leads to destructive interference with zero intensity.
• General Phase Difference \(\phi\): Any other value causes varying degrees of intensity as waves partially reinforce or cancel each other.

By adjusting the phase difference, one can control the resultant patterns and better understand the experimental results. It is this manipulation that allows scientists to fine-tune and predict wave behaviors.

Intensity Ratio

The intensity ratio is used to quantify the differences in brightness between the brightest and darkest fringes in an interference pattern. It is given by the formula \(\frac{I_{\mathrm{max}}}{I_{\mathrm{min}}} = \frac{(\sqrt{I_{1}} + \sqrt{I_{2}})^{2}}{(\sqrt{I_{1}} - \sqrt{I_{2}})^{2}}\).

In this equation:

• \(I_{\mathrm{max}}\) is the intensity where the waves constructively interfere.
• \(I_{\mathrm{min}}\) is the intensity where they destructively interfere.
• \(I_1\) and \(I_2\) are the intensities of the earlier mentioned coherent sources.

By measuring the amplitude relationship between two sources, the intensity ratio provides insight into the extent of constructive or destructive interference. It allows for a pragmatic understanding of how combined wave sources contribute to the observed intensity patterns.