Question

Two coherent monochromatic light beams if intensity $4l$ and $9l$ are superimposed. The maximum and minimum intensities of the resultant beam are
A) $3l$ and $2l$
B) $9l$ and $5l$
C) $16l$ and $3l$
D) $25l$ and $l$

Answer 1

Use the principle of superposition to find the maximum and minimum intensities of the resultant beam. Coherent beams mean that the phase angle between them is zero. Phase angle may refer to the angular displacement of a sinusoid from a reference point or time.

Complete step-by-step answer:
Here, it is given in the question that the sources are coherent which means that the phase angle between them is zero.
According to the principle of superposition of waves,
Maximum intensity, ${I_{\max }} = {\left( {\sqrt {{I_1}} + \sqrt {{I_2}} } \right)^2}$
Where ${I_1}$and ${I_2}$are the intensities of the first and second beam respectively.
${I_{\max }} = {\left( {\sqrt {4l} + \sqrt {9l} } \right)^2}$
${I_{\max }} = {\left( {2\sqrt l + 3\sqrt l } \right)^2}$
${I_{\max }} = 25l$
Similarly,
Minimum intensity, ${I_{\min }} = {\left( {\sqrt {{I_1}} - \sqrt {{I_2}} } \right)^2}$
${I_{\min }} = {\left( {\sqrt {4l} - \sqrt {9l} } \right)^2}$
${I_{\min }} = {\left( {2\sqrt l - 3\sqrt l } \right)^2}$
${I_{\min }} = l$

The maximum and minimum intensities of the resultant beam are $25l$ and $l$. So, Option (D) is correct.

Additional information:
The principle of superposition of waves describes how the individual waveforms can be algebraically added to determine the net waveform. Waveform tells about the overall motion of the wave. It does not tell about individual particles of the wave.
Constructive and destructive interference occurs due to the principle of superposition. According to this principle, when several waves of the same type meet at a point, the resultant displacement at that point is the sum of the displacements due to each of the incident waves. The main difference between constructive and destructive interference is that constructive interference occurs when the displacements of the waves that meet are in the same direction, whereas destructive interference occurs when displacements of the waves that meet are in the opposite directions.

Note:
The waves are said to be incoherent if they do not have a constant phase difference. These sources will produce light with random and frequent changes of phase between the photons. If the sources are incoherent I stead of coherent then we can directly add the intensities to find the resultant intensity of the resultant beam.