Question

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

Answer 1

When two coherent monochromatic light beams of intensities ${I_1}$ and ${I_2}$ are superposed, the maximum intensity is calculated by square of sum of both intensities ${(\sqrt {I_1} + \;\sqrt {I_2} )^2}$ . When two coherent monochromatic light beams of intensities ${I_1}$ and ${I_2}$ are superposed, the minimum intensity is calculated by square of difference of both intensities ${(\sqrt {I_1} - \sqrt {I_2} )^2}$

Complete step by step answer:
Given : ${{I_1} = I}$ and ${I_2} = {\text{ }}4I$
When two coherent monochromatic light beams of intensities ${\text{I}}$ and ${\text{4I}}$ are superposed, the maximum intensity is calculated by square of sum of both intensities
$I{}_{\max } = {(\sqrt {I_1} + \;\sqrt {I_2} )^2}$
$\Rightarrow I{}_{\max } = {(\sqrt {I} + \;\sqrt {4I} )^2}$
$\Rightarrow {I_{\max }} = {(\sqrt I )^2} + {(\sqrt {4I} )^2} + 2 \times (\sqrt I ) \times (\sqrt {4I} )$
$\Rightarrow {I_{\max }} = {\text{ }}I{\text{ }} + {\text{ }}4I{\text{ }} + {\text{ }}4I$
$\Rightarrow {I_{\max }}\; = {\text{ }}9I$
When two coherent monochromatic light beams of intensities ${\text{I}}$ and ${\text{4I}}$ are superposed, the minimum intensity is calculated by square of difference of both intensities
$I{}_{\min } = {(\sqrt {I_1} - \;\sqrt {I_2} )^2}$
$\Rightarrow I{}_{\min } = {(\sqrt {I} - \;\sqrt {4I} )^2}$
$\Rightarrow {I_{\min }} = {(\sqrt I )^2} + {(\sqrt {4I} )^2} - 2 \times (\sqrt I ) \times (\sqrt {4I} )$
$\Rightarrow {I_{\min }}\; = {\text{ }}I{\text{ }} + {\text{ }}4I{\text{ }} - {\text{ }}4I$
$\Rightarrow {I_{\min }}\;{\text{ }} = {\text{ }}I$
The maximum and minimum possible resulting intensities are : ${{\text{I}}_{{\text{max}}}}{\text{= 9I}}$ and ${{\text{I}}_{{\text{min}}}}{\text{ = I}}$ respectively.

Hence, option (D) is the correct answer.

Note: When two coherent monochromatic light beams of intensities ${I_1}$ and ${I_2}$ are superposed, the maximum intensity is calculated by square of sum of both intensities ${(\sqrt {I_1} + \;\sqrt {I_2} )^2}$. When two coherent monochromatic light beams of intensities ${I_1}$ and ${I_2}$ are superposed, the minimum intensity is calculated by square of difference of both intensities ${(\sqrt {I_1} - \sqrt {I_2} )^2}$.