Question

The light waves from two coherent sources of same intensity interfere each other. Then what will be maxima intensity when minimum intensity is zero ? (A) \(4 \mathrm{I}\) (B) I (C) \(4 \mathrm{I}^{2}\) (D) \(\mathrm{I}^{2}\)

Short answer

The maximum intensity when the minimum intensity is zero is \(4I\).

Step-by-step solution

Understanding Constructive and Destructive Interference

When two waves interfere, the resulting intensity depends on the phase difference between the waves. When the phase difference is an even multiple of pi, the interference is constructive, and the waves add up, increasing the intensity at that point. When the phase difference is an odd multiple of pi, the interference is destructive, and the waves cancel each other out, lowering the intensity at that point. Given that the minimum intensity is zero, we have the case of complete destructive interference between the two waves. Now, we need to find the maximum intensity resulting from the constructive interference between them.

Calculating the Maximum Intensity

In the case of constructive interference, both the waves add up, and their intensities get added. Let the intensity of each wave be I. Then, the maximum intensity will be the sum of the intensities of both waves. Maximum Intensity = Intensity of Wave 1 + Intensity of Wave 2

Determine the Maximum Intensity using the given information

Since both waves have the same intensity I, we can rewrite the equation as: Maximum Intensity = I + I Thus, the maximum intensity will be: Maximum Intensity = 2I However, when the intensity at a point is the result of the interference of two waves, the maximum possible intensity can be up to four times the individual intensity. This happens when the amplitude of the resulting wave doubles due to the perfect constructive interference between the two waves, as the square of amplitude of the wave is directly proportional to the intensity. Therefore, the maximum possible intensity is: Maximum Intensity = 4I This matches option (A). So, the maximum intensity when the minimum intensity is zero is \(4I\).

Key concepts

Constructive Interference

When two light waves meet and their crests align perfectly, they undergo constructive interference. This occurs when their phase difference is an even multiple of \( \pi \), meaning they meet "in phase." In this case, the amplitudes of the waves add up, leading to a higher total intensity.

• Imagine two identical waves overlapping.
• Their heights (amplitudes) combine, resulting in a wave with doubled amplitude.

Since intensity is proportional to the square of amplitude, the combined wave has an intensity of four times that of an individual wave. So, for two colliding coherent light waves of intensity \( I \), the highest possible intensity from constructive interference would be \( 4I \).
This is why the answer to the exercise is 4I. It demonstrates the full potential of waves reinforcing each other when perfectly in phase.

Destructive Interference

Destructive interference is the opposite of constructive interference. It happens when two waves meet "out of phase," meaning their troughs and crests are misaligned. Specifically, their phase difference is an odd multiple of \( \pi \), causing the waves to cancel each other out.

• Think of one wave's crest overlapping with another wave's trough.
• This overlap causes the waves to reduce each other's effect.

If both waves have equal intensity and are perfectly out of phase, they will completely cancel each other. This leaves zero intensity as the result. Thus, in the problem, when they mention minimum intensity is zero, it signifies complete destructive interference. There's no light at those points, illustrating the profound nullifying effect of opposite wave peaks meeting.

Coherent Sources

Coherent sources are crucial for both constructive and destructive interference. Coherent sources are light sources that emit waves with a constant phase difference, frequency, and identical wavelength.

• Picture two lasers that emit light in lockstep with each other.
• Their emissions maintain a constant relative position.

This consistent phase relationship is necessary for interference patterns to develop. The waves must maintain their synchronization as they overlap. Without coherence, any pattern of bright or dark interference fringes would dissolve into random flickers of light. In essence, coherent sources allow for stable and predictable interference, just as we examined in the exercise when calculating maxima and minima of intensity.