Problem 1 - Master Resource Book in JEE Main Physics

Question

The ratio of intensities of two waves is \(9: 1\). They are producing interference. The ratio of maximum and minimum intensities will be (a) \(10: 8\) (b) \(9: 1\) (c) \(4: 1\) (d) \(12: 1\)

Step-by-Step Solution

Answer

The ratio of maximum to minimum intensities is \(4:1\). (Option c)

Step 1: Determine the amplitude relationship

The intensity of a wave is proportional to the square of its amplitude. Thus, if the intensity ratio is 9:1, the amplitude ratio can be found by taking the square root of each term. This gives us an amplitude ratio of \(\sqrt{9}:\sqrt{1} = 3:1\).

Step 2: Calculate the maximum intensity

When two waves interfere constructively, the maximum intensity is given by the formula \((a_1 + a_2)^2\), where \(a_1\) and \(a_2\) are the amplitudes of the waves. With amplitudes 3 and 1, the maximum intensity is \((3 + 1)^2 = 16\).

Step 3: Calculate the minimum intensity

When two waves interfere destructively, the minimum intensity is given by the formula \((a_1 - a_2)^2\). With amplitudes 3 and 1, the minimum intensity is \((3 - 1)^2 = 4\).

Step 4: Determine the ratio of maximum to minimum intensity

The ratio of maximum intensity to minimum intensity is then \(16:4\), which simplifies to \(4:1\). Therefore, the ratio of maximum to minimum intensities is \(4:1\).

Key Concepts

Intensity and Amplitude Relationship

In physics, understanding the relationship between intensity and amplitude is crucial when studying wave phenomena, especially in wave interference. The intensity of a wave is closely linked to its amplitude. Specifically, it is proportional to the square of the amplitude. This means that if you know the amplitude, you can find the intensity by squaring it. For instance, if the amplitude of one wave is three times larger than another, its intensity is nine times greater, as \[ I \propto A^2 \]. This square relationship shows how much more energy a wave can carry when its amplitude increases.
In the context of the problem where the intensity ratio is 9:1, we can extrapolate that the amplitude ratio must be the square root of each intensity term. Thus, the amplitude ratio is \( \sqrt{9} : \sqrt{1} = 3:1 \). This transformation highlights how even a small change in amplitude can result in a significant change in intensity.

Constructive Interference

When waves meet each other, they can interfere in a way that results in a more powerful wave. This is known as constructive interference. Constructive interference occurs when the crest (the highest point) of one wave aligns with the crest of another wave. In this situation, the amplitudes add up, leading to a wave that has greater intensity.
For example, if two waves with amplitudes of 3 and 1 meet constructively, the combined amplitude is the sum of the two: \( a_1 + a_2 = 3 + 1 = 4 \). The intensity of the resultant wave can then be calculated using the formula for intensity, \((a_1 + a_2)^2\). Thus, the maximum intensity for these two waves is \( (3 + 1)^2 = 16 \). This maximum intensity shows how waves can combine to form a wave of increased energy.

Destructive Interference

Destructive interference is a phenomenon where two waves intersect in such a manner that they effectively cancel each other out. This occurs when the crest of one wave meets the trough (the lowest point) of another wave, which lowers the overall amplitude of the resulting wave.
In our example, waves with amplitudes of 3 and 1 experiencing destructive interference would lead to a reduced amplitude, calculated as the difference between individual amplitudes: \( a_1 - a_2 = 3 - 1 = 2 \). The formula used to calculate the minimum intensity, in this case, is \((a_1 - a_2)^2\). Consequently, the minimum intensity is equal to \( (3 - 1)^2 = 4 \). This resulting lower intensity illustrates how waves can interfere to decrease overall energy, making some portions quieter and less visible.