Problem 52 - Master Resource Book in JEE Main Physics

Question

Two waves having the intensities in the ratio \(9: 1\) produce interference. The ratio of maximum to minimum intensity is equal to (a) \(10: 8\) (b) \(9: 1\) (c) \(4: 1\) (d) \(2: 1\)

Step-by-Step Solution

Answer

The ratio of maximum to minimum intensity is 4:1.

Step 1: Understand the Intensity Ratio

The intensity ratio of the two interfering waves is given as 9:1. Let the intensities of the waves be \(I_1\) and \(I_2\) such that \(\frac{I_1}{I_2} = 9\). This means \(I_1 = 9I_2\).

Step 2: Recall the Formula for Interference Intensities

In interference patterns, the maximum intensity \(I_{\text{max}}\) is given by \((\sqrt{I_1} + \sqrt{I_2})^2\) and the minimum intensity \(I_{\text{min}}\) is given by \((\sqrt{I_1} - \sqrt{I_2})^2\).

Step 3: Calculate Maximum Intensity

Substitute \(I_1 = 9I_2\) into the maximum intensity formula: \(I_{\text{max}} = (\sqrt{9I_2} + \sqrt{I_2})^2 = (3\sqrt{I_2} + \sqrt{I_2})^2 = (4\sqrt{I_2})^2 = 16I_2\).

Step 4: Calculate Minimum Intensity

Substitute \(I_1 = 9I_2\) into the minimum intensity formula: \(I_{\text{min}} = (\sqrt{9I_2} - \sqrt{I_2})^2 = (3\sqrt{I_2} - \sqrt{I_2})^2 = (2\sqrt{I_2})^2 = 4I_2\).

Step 5: Find the Ratio of Maximum to Minimum Intensity

The ratio of the maximum to the minimum intensity is \(\frac{I_{\text{max}}}{I_{\text{min}}} = \frac{16I_2}{4I_2} = 4:1\).

Key Concepts

Understanding Intensity Ratio

The intensity ratio in wave interference tells us how the energy of the two waves compares. When we say that the intensity ratio is 9:1, it means that the energy carried by the first wave is nine times greater than that of the second wave.
For such an intensity ratio, let's define the intensity of the first wave as \(I_1\) and of the second wave as \(I_2\). The ratio \( \frac{I_1}{I_2} = 9 \), implies that \(I_1 = 9I_2\).
In simple terms, one wave is much stronger, representing the 9 parts of the total intensity, while the other wave contributes just 1 part.

Calculating Maximum Intensity

In an interference pattern, maximum intensity occurs when the waves perfectly combine or sync together. The constructive interference creates bright spots, represented by high intensity values.
The maximum intensity \( I_{\text{max}} \) of two waves in interference is calculated using the formula:

• \( I_{\text{max}} = (\sqrt{I_1} + \sqrt{I_2})^2 \).

To find this in the given problem where \(I_1 = 9I_2\), the calculation follows:

• \( I_{\text{max}} = (\sqrt{9I_2} + \sqrt{I_2})^2 = (3\sqrt{I_2} + \sqrt{I_2})^2 = (4\sqrt{I_2})^2 = 16I_2 \).

This value represents the strongest point in the interference pattern.

Finding Minimum Intensity

Minimum intensity happens when waves partially cancel each other out, resulting in dark spots in the pattern. This phenomenon is known as destructive interference.
The formula for calculating minimum intensity \( I_{\text{min}} \) is:

• \( I_{\text{min}} = (\sqrt{I_1} - \sqrt{I_2})^2 \).

In our exercise, substituting \(I_1 = 9I_2\) gives:

• \( I_{\text{min}} = (\sqrt{9I_2} - \sqrt{I_2})^2 = (3\sqrt{I_2} - \sqrt{I_2})^2 = (2\sqrt{I_2})^2 = 4I_2 \).

This calculation indicates the weakest point of intensity within the interference pattern.

Understanding Wave Intensity

Wave intensity is a measure of how much energy the wave is transmitting per unit of area. It represents how strong or weak a wave is at any given point. In physics, wave intensity is essential for understanding wave interactions, particularly in interference patterns.
In the context of the interference problem, intensity reflects the power that waves distribute across the interference pattern.
The importance of calculating maximum and minimum intensities is evident here as they help identify where the interference is most constructive or destructive. This is critical for applications like observing wave behavior in light patterns or sound waves.