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Physics Question on Wave optics

Two coherent sources of intensity ratio β\beta interfere. Then the value of (Imax⁡−Imin⁡)/(Imax⁡+Imin⁡)(I_{\max }-I_{\min })/(I_{\max }+I_{\min }) is:

A

1+ββ\frac{1+\beta}{\sqrt{\beta}}

B

(1+ββ)\sqrt{\left(\frac{1+\beta}{\beta}\right)}

C

1+β2β\frac{1+\beta}{2\beta}

D

2β1+β\frac{2 \sqrt{\beta}}{1+\beta}

Answer

2β1+β\frac{2 \sqrt{\beta}}{1+\beta}

Explanation

Solution

We know I∝a2I \propto a^{2} or a∝Ia \propto \sqrt{I} ∴a1a2=(I1I2)\therefore \frac{a_{1}}{a_{2}}=\sqrt{\left(\frac{I_{1}}{I_{2}}\right)} So, Imax⁡Imin⁡=(a1+a2)2(a1−a2)2\frac{I_{\max }}{I_{\min }}=\frac{\left(a_{1}+a_{2}\right)^{2}}{\left(a_{1}-a_{2}\right)^{2}} =(a1+a2)(a2−a1)2=(1+β)2(1−β)2=\frac{\left(a_{1}+a_{2}\right)}{\left(a_{2}-a_{1}\right)^{2}}=\frac{(1+\sqrt{\beta})^{2}}{(1-\sqrt{\beta})^{2}} Applying componendo and dividendo Imax⁡+Imin⁡Imax⁡−Imin⁡=(1+β)2+(1−β)2(1+β)2−(1−β)2\frac{I_{\max }+I_{\min }}{I_{\max }-I_{\min }}=\frac{(1+\sqrt{\beta})^{2}+(1-\sqrt{\beta})^{2}}{(1+\sqrt{\beta})^{2}-(1-\sqrt{\beta})^{2}} or =2+2β4β=\frac{2+2 \beta}{4 \sqrt{\beta}} or Imax⁡−Imin⁡Imax⁡+Imin⁡=2β1+β\frac{I_{\max }-I_{\min }}{I_{\max }+I_{\min }}=\frac{2 \sqrt{\beta}}{1+\beta}