Question

Consider two coherent, monochromatic (wavelength $\lambda $ ) sources, $S_{1}$ and $S_{2}$ , separated by a distance $d$ . The ratio of intensities of $S_{1}$ and that of $S_{2}$ (which is responsible for interference at point $P$ , where detector is located) is $4$ . The distance of point $P$ from $S_{1}$ is (if the resultant intensity at point $P$ is equal to $\frac{9}{4}$ times of intensity of $S_{1}$ ) (Given: $\angle S_{2}S_{1}P=90^\circ $ , $d>0$ and $n$ is a positive integer)

• A. $\frac{d^{2} - n^{2} \lambda ^{2}}{2 n \lambda }$
• B. $\frac{d^{2} + n^{2} \lambda ^{2}}{2 n \lambda }$
• C. $\frac{n \lambda d}{\sqrt{d^{2} - n^{2} \lambda ^{2}}}$
• D. $\frac{2 \, n \, \lambda \, d}{\sqrt{d^{2} - n^{2} \lambda ^{2}}}$

Answer 1

Answer: (A)

$\frac{9}{4}$ intensity of $S_{1}=9$ times of intensity $S_{2}$
Intensity of maxima = $\left(\sqrt{I_{1}} + \sqrt{I_{2}}\right)^{2}=9I_{2}$
$\Rightarrow S_{2}P-S_{1}P=n\lambda $ for maxima
$\sqrt{d^{2} + x^{2}}-x=n\lambda \, x=\frac{d^{2} - n^{2} \lambda ^{2}}{2 n \lambda }$