Question

The ratio of the intensities at maxima to minima in Young’s double-slit experiment is \( 25 : 9 \). Calculate the ratio of intensities of the interfering waves.

Hint:

Use the identity: \[ \frac{I_{\text{max}}}{I_{\text{min}}} = \left( \frac{\sqrt{I_1} + \sqrt{I_2}}{\sqrt{I_1} - \sqrt{I_2}} \right)^2 \] Then apply componendo and dividendo to simplify the root ratio.

Answer 1

In Young's double-slit experiment, the ratio of maximum to minimum intensity is expressed as: \[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{(\sqrt{I_1} + \sqrt{I_2})^2}{(\sqrt{I_1} - \sqrt{I_2})^2} \] It is given that: \[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{25}{9} \] Taking the square root of both sides yields: \[ \frac{\sqrt{I_1} + \sqrt{I_2}}{\sqrt{I_1} - \sqrt{I_2}} = \frac{5}{3} \] Applying the componendo and dividendo rule: \[ \frac{\sqrt{I_1}}{\sqrt{I_2}} = \frac{5 + 3}{5 - 3} = \frac{8}{2} = 4 \] Squaring this result gives the ratio of intensities: \[ \frac{I_1}{I_2} = 4^2 = \boxed{16:1} \]