Question

Two coherent monochromatic beams of intensities $I$ and $4I$ respectively are superposed. The ration of the maximum and minimum intensities in the resulting pattern is

• A. $5:3$
• B. $3:1$
• C. $4:1$
• D. $9:1$

Answer 1

The correct option is D

$9:1$

Step 1: Given data

Intensities of the two monochromatic beams are $I_1=4I$ and $I_2=I$ respectively.

Step 2: To find

The ratio of maximum and minimum intensities.

Step 3: Formula used

Maximum intensity,

$I_{max}={\left(\sqrt{I_1}+\sqrt{I_2}\right)}^2$

Minimum intensity,

$I_{min}={\left(\sqrt{I_1}-\sqrt{I_2}\right)}^2$

Step 4: Calculation

Ratio of maximum and minimum intensities,

$$\begin{gathered} \frac{I_{max}}{I_{min}}=\frac{{\left(\sqrt{I_1}+\sqrt{I_2}\right)}^2}{{\left(\sqrt{I_1}-\sqrt{I_2}\right)}^2} \\ \Rightarrow \frac{I_{max}}{I_{min}}=\frac{{\left(\sqrt{4I}+\sqrt{I}\right)}^2}{{\left(\sqrt{4I}-\sqrt{I}\right)}^2} \\ \Rightarrow \frac{I_{max}}{I_{min}}={\left(\frac{\sqrt{4I}+\sqrt{I}}{\sqrt{4I}-\sqrt{I}}\right)}^2 \\ \Rightarrow \frac{I_{max}}{I_{min}}={\left(\frac{\left(\sqrt{4I}+\sqrt{I}\right)\left(\sqrt{4I}+\sqrt{I}\right)}{\left(\sqrt{4I}-\sqrt{I}\right)\left(\sqrt{4I}+\sqrt{I}\right)}\right)}^2 \\ \Rightarrow \frac{I_{max}}{I_{min}}={\left(\frac{{\left(\sqrt{4I}+\sqrt{I}\right)}^2}{{\left(\sqrt{4I}\right)}^2-{\left(\sqrt{I}\right)}^2}\right)}^2 \\ \Rightarrow \frac{I_{max}}{I_{min}}={\left(\frac{I+4I+\left(2\sqrt{I}\sqrt{4I}\right)}{4I-I}\right)}^2 \\ \Rightarrow \frac{I_{max}}{I_{min}}={\left(\frac{5I+\left(2\times 2I\right)}{3I}\right)}^2 \\ \Rightarrow \frac{I_{max}}{I_{min}}={\left(\frac{9I}{3I}\right)}^2 \\ \Rightarrow \frac{I_{max}}{I_{min}}={\left(\frac{3}{1}\right)}^2 \\ \Rightarrow \frac{I_{max}}{I_{min}}=\frac{9}{1} \end{gathered}$$

Therefore, the ratio of the maximum and minimum intensities in the resulting pattern is $9:1$.

Hence, option D is correct.