Question

Two coherent monochromatic light beams of intensities \(I\) and \(4I\) are superposed. What are the maximum and minimum possible intensities in the resulting beam?

• A. \(I_{\max}=2I,\ I_{\min}=I\)
• B. \(I_{\max}=4I,\ I_{\min}=I\)
• C. \(I_{\max}=9I,\ I_{\min}=3I\)
• D. \(I_{\max}=9I,\ I_{\min}=I\)

Hint:

Remember: \[ I_{\max} = (\sqrt{I_1}+\sqrt{I_2})^2 \] \[ I_{\min} = (\sqrt{I_1}-\sqrt{I_2})^2 \] for two coherent interfering sources.

Answer 1

The Correct Option is D

Concept: For interference of two coherent waves, \[ I_{\max} = (\sqrt{I_1}+\sqrt{I_2})^2 \] \[ I_{\min} = (\sqrt{I_1}-\sqrt{I_2})^2 \]

Step 1: Identify the intensities. \[ I_1=I, \qquad I_2=4I \] Hence, \[ \sqrt{I_1}=\sqrt I, \qquad \sqrt{I_2}=2\sqrt I \]

Step 2: Calculate the maximum intensity. \[ I_{\max} = (\sqrt I+2\sqrt I)^2 \] \[ = (3\sqrt I)^2 \] \[ = 9I \]

Step 3: Calculate the minimum intensity. \[ I_{\min} = (2\sqrt I-\sqrt I)^2 \] \[ = (\sqrt I)^2 \] \[ = I \] \[\begin{aligned} \boxed{I_{\max}=9I,\quad I_{\min}=I} \end{aligned}\] Hence, option \(\mathbf{(D)}\) is correct.