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# Two waves of intensities $I$ and $4I$ produce interference. Then the intensity of constructive and destructive interferences respectively is

Last updated date: 24th Jul 2024
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Hint: We remember that any light wave's intensity is directly proportional to the amplitude square. The waves are coherent if they own the same wavelength, frequency, and fixed phase difference. For Example, sound waves from two loudspeakers by the same audio oscillator produce coherent waves.

Complete step-by-step solution:
Given: $I_{1} = I$ and $I_{2} = 4I$
For the interference, the sources must be coherent. The interference of light waves is described as the modification in the division of light energy when two or more waves combine.
The resultant intensity of two coherent sources is: -
$I_{R} = I_{1} + I_{2} + 2 \sqrt{ I_{1} I_{2}} cos \phi$
There are two kinds of Interference: - destructive interference and constructive interference.
In constructive interference, the wave's amplitude combines when the light waves are at an identical place and time. In this state, the waves are in the same phase.
$cos \phi = 1$
$I_{R} = \left( \sqrt{ I_{1} } + \sqrt{ I_{2} } \right)^{2}$
$\implies I_{R} = \left( \sqrt{ I} + \sqrt{ 4I} \right)^{2}$
$\implies I_{R} = 9I$
In destructive interference, the waves stay out of phase. So, when the waves join together, the resultant wave equals zero.
$cos \phi = -1$
$I_{R} = \left( \sqrt{ I_{1} } - \sqrt{ I_{2} } \right)^{2}$
$\implies I_{R} = \left( \sqrt{ I} - \sqrt{ 4I} \right)^{2}$
$\implies I_{R} = I$
Hence, the intensity of constructive and destructive interferences is $9I$ and $I$ respectively.

Note: There are two ways to generate coherent sources. The first method is by the distribution of wavefront. The wavefront is split into two parts or more parts by utilizing lenses, prisms, and mirrors. A wavefront is a surface in which an optical wave contains a constant phase. It can crest or the trough of the same wave. The second method is by distributing the wave's amplitude by two parts or more components through biased reflection or refraction. These divided parts may move through different routes and finally join collectively to produce interference.