Question

Two coherent sources of intensity ratio $\beta $ interfere. Then the value of $\left(I_{max} - I_{min}\right)/\left(I_{max} + I_{min}\right)$ is :-

• A. $\frac{1 + \beta }{\sqrt{\beta }}$
• B. $\sqrt{\frac{1 + \beta }{\beta }}$
• C. $\frac{1 + \beta }{2 \sqrt{\beta }}$
• D. $\frac{2 \sqrt{\beta }}{1 + \beta }$

Answer 1

Answer: (D)

Let intensity of be the two coherent sources are $I_{1}\text{and}I_{2}$ .
Then, $\beta =\frac{I_{1}}{I_{2}}$
Now, $\frac{I_{max}}{I_{min}}=\frac{\left(\sqrt{I_{1}} + \sqrt{I_{2}}\right)^{2}}{\left(\sqrt{I_{1}} - \sqrt{I_{2}}\right)^{2}}=\frac{\left(\sqrt{\frac{I_{1}}{I_{2}}} + 1\right)^{2}}{\left(\sqrt{\frac{I_{1}}{I_{2}}} - 1\right)^{2}}=\frac{\left(\sqrt{\beta } + 1\right) ^{2}}{\left(\sqrt{\beta } - 1\right) ^{2}}$
$\Rightarrow \frac{I_{max}}{I_{min}}=\frac{\beta + 1 + 2 \sqrt{\beta }}{\beta + 1 - 2 \sqrt{\beta }}$
Now, $\frac{I_{max} - I_{min}}{I_{max} + I_{min}}=\frac{\frac{I_{max}}{I_{min}} - 1}{\frac{I_{max}}{I_{min}} + 1}=\frac{2 \sqrt{\beta }}{1 + \beta }$