The interference pattern is obtained with two coherent light sources of intensity ratio . In the interference pattern, the ratio will be [2016]

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Solution

Q.
The interference pattern is obtained with two coherent light sources of intensity ratio $n$ . In the interference pattern, the ratio $\frac{I_{max} - I_{min}}{I_{max} + I_{min}}$ will be [2016]

1 $\frac{\sqrt{n}}{n + 1}$

2 $\frac{2 \sqrt{n}}{n + 1}$

3 $\frac{\sqrt{n}}{{(n + 1)}^{2}}$

4 $\frac{2 \sqrt{n}}{{(n + 1)}^{2}}$

Ans.
(2)

Here, $\frac{I_{1}}{I_{2}} = n$

$\frac{I_{max}}{I_{min}} = {(\frac{\sqrt{I_{1}} + \sqrt{I_{2}}}{\sqrt{I_{1}} - \sqrt{I_{2}}})}^{2} = {(\frac{\sqrt{I_{1} / I_{2}} + 1}{\sqrt{I_{1} / I_{2}} - 1})}^{2} = {(\frac{\sqrt{n} + 1}{\sqrt{n} - 1})}^{2}$

$\frac{I_{max} - I_{min}}{I_{max} + I_{min}} = \frac{\frac{I_{max}}{I_{min}} - 1}{\frac{I_{max}}{I_{min}} + 1}$

$= \frac{{(\frac{\sqrt{n} + 1}{\sqrt{n} - 1})}^{2} - 1}{{(\frac{\sqrt{n} + 1}{\sqrt{n} - 1})}^{2} + 1} = \frac{{(\sqrt{n} + 1)}^{2} - {(\sqrt{n} - 1)}^{2}}{{(\sqrt{n} + 1)}^{2} + {(\sqrt{n} - 1)}^{2}}$ $= \frac{4 \sqrt{n}}{2 (n + 1)} = \frac{2 \sqrt{n}}{n + 1}$

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