Question

Two coherent monochromatic light beams of intensities ${\text{I}}$ and $4{\text{I}}$ are superposed. The maximum and minimum possible resulting intensities are:
A $5{\text{I}}$ and $3{\text{I}}$
B $9{\text{I}}$ and $3{\text{I}}$
C $4{\text{I}}$ and ${\text{I}}$
D $9{\text{I}}$ and ${\text{I}}$

Answer 1

Hint: Monochromatic light is the light where the optical spectrum only and only one optical frequency. It has zero bandwidth and instantaneous frequency. The monochromatic signifies that it only possesses a single color. Lasers are the primary source of monochromatic light. This makes for very small or narrow bandwidth; it creates very sharply defined interference bands.
Coherent light ensures that all the photons must be in the same phase; this shows that for a single frequency all of the peaks are equally spaced.


Complete step by step solution:
Given that, the two coherent monochromatic light beams of intensities ${\text{I}}$ and $4{\text{I}}$ .
Let, The first monochromatic light beam of intensity ${\text{I}}$ is ${{\text{I}}_1}$
The second monochromatic light beam of intensity $4{\text{I}}$ is ${{\text{I}}_2}$
As we know that,
The maximum intensities ${{\text{I}}_{\max }} = (\sqrt {{{\text{I}}_1}} + {\sqrt {{{\text{I}}_2})} ^2}$
$ = (\sqrt {\text{I}} + {\sqrt {4{\text{I}})} ^2}$
By using formula;
$ = ({\sqrt {{\text{I}})} ^2} + ({\sqrt {4{\text{I}})} ^2} + 2 \times \sqrt I \times \sqrt {4{\text{I}}} $
$ = 9{\text{I}}$
The minimum intensities ${{\text{I}}_{\min }} = (\sqrt {{{\text{I}}_1}} - {\sqrt {{{\text{I}}_2})} ^2}$
$ = (\sqrt {\text{I}} - {\sqrt {4{\text{I}})} ^2}$
By using formula;
\[ = {\left( {\sqrt 1 } \right)^2} + {\left( {\sqrt {41} } \right)^2} - 2 \times \sqrt I \times \sqrt {4{\text{I}}} = 1\]
The maximum and minimum possible resulting intensities are $9{\text{I}}$, and${\text{I}}$.
Hence, the correct option is D.


Note: The example of monochromatic and coherent sources of light is laser beam. Two waves are said to be coherent when their phase difference is constant and their frequency must be the same. Monochromatic sources of light will emit monochromatic light.