Question

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

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

Answer 1

Answer: (D)

Let intensity of be the two coherent sources are $I_1\text{and}{I}_2$ .
Then, $β=\frac{I_1}{I_2}$
Now, $\frac{I_{max}}{I_{min}}=\frac{{\left(\sqrt{I_1}+\sqrt{I_2}\right)}^2}{{\left(\sqrt{I_1}−\sqrt{I_2}\right)}^2}=\frac{{\left(\sqrt{\frac{I_1}{I_2}}+1\right)}^2}{{\left(\sqrt{\frac{I_1}{I_2}}−1\right)}^2}=\frac{{\left(\sqrt{β}+1\right)}^2}{{\left(\sqrt{β}−1\right)}^2}$
$⇒\frac{I_{max}}{I_{min}}=\frac{β+1+2\sqrt{β}}{β+1−2\sqrt{β}}$
Now, $\frac{I_{max}−I_{min}}{I_{max}+I_{min}}=\frac{\frac{I_{max}}{I_{min}}−1}{\frac{I_{max}}{I_{min}}+1}=\frac{2\sqrt{β}}{1+β}$