Question

Two coherent monochromatic light beams of intensities ratio 1 : 4 are superposed. The ratio of maximum and minimum intensities in the resulting beam will be

• A. 9:01
• B. 5:03
• C. 25:09:00
• D. 9:25

Answer 1

Answer: (A)

9:01

Answer 2

Given, $I_1 : I_2 = 1 : 4$
As, we know
$I_{max} =\left(\sqrt{I_{1}} +\sqrt{I_{2}}\right)^{2}$
and $I_{min} =\left(\sqrt{I_{1}} -\sqrt{I_{2}}\right)^{2}$
Hence, the ratio
$\frac{I_{max}}{I_{min}} =\left[\frac{\sqrt{I_{1}} +\sqrt{I_{2}}}{\sqrt{I_{1}} -\sqrt{I_{2}}}\right] = \left[\frac{\sqrt{\frac{I_{1}}{I_{2}}}+1}{\sqrt{\frac{I_{1}}{I_{2}}}-1}\right]^{2}$
$\Rightarrow \frac{I_{max}}{I_{min}} = \left[\frac{\sqrt{\frac{1}{4}}+1}{\sqrt{\frac{1}{4}}-1}\right] =\left[\frac{\frac{1+2}{2}}{\frac{1-2}{2}}\right]^{2} = \left[\frac{3}{-1}\right]^{2} = 9 : 1$
$\Rightarrow \frac{I_{max}}{I_{min}} = \frac{9}{1}$
Hence, the ratio of maximum and minimum intensities in the resulting beam is $9 : 1$.