Two wave having the intensities in the ratio of 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

To determine the ratio of maximum to minimum intensity produced by the interference of two waves with intensities in the ratio of 9:1, we can use the principle of superposition along with the formula for interference. Let’s break it down step-by-step.

Understanding Intensities and Their Ratio

The intensity of a wave is a measure of the power per unit area. When two waves interfere, the resulting intensity can be calculated based on their individual intensities. In this case, we have two waves with intensities \( I_1 \) and \( I_2 \), where \( I_1:I_2 = 9:1 \). This means we can express them as:

• Let \( I_1 = 9I \)
• Let \( I_2 = I \)

Calculating Maximum and Minimum Intensities

When these two waves interfere constructively (in phase), the maximum intensity \( I_{max} \) can be calculated using the formula:

I_{max} = I_1 + I_2

Substituting our values:

I_{max} = 9I + I = 10I

On the other hand, when the waves interfere destructively (out of phase), the minimum intensity \( I_{min} \) is given by:

I_{min} = |I_1 - I_2|

Again, substituting our values:

I_{min} = |9I - I| = |8I| = 8I

Finding the Intensity Ratio

Now that we have both maximum and minimum intensities, we can find the ratio:

Ratio = \frac{I_{max}}{I_{min}} = \frac{10I}{8I} = \frac{10}{8} = \frac{5}{4}

Final Ratio Interpretation

The ratio of maximum to minimum intensity simplifies to 5:4. However, since this is not one of the provided options, let’s review the choices given:

• (a) 10 : 8
• (b) 9 : 1
• (c) 4 : 1
• (d) 2 : 1

To express 5:4 in a form similar to one of the options, we can multiply both parts of the ratio by 2.5 to yield 10:8, which corresponds to option (a).

Conclusion

Thus, the correct answer to the question is (a) 10:8. This illustrates how interference can lead to varying intensity levels based on the initial intensities of the waves involved.