Consider two coherent monochromatic (wavelength $λ$ ) sources $S_{1}$ and $S_{2}$ separated by a distance $d$ . The ratio of intensity of $S_{1}$ and that of $S_{2}$ at point $P$ is 4. The distance of $P$ from $S_{1}$ if the resultant intensity at point $P$ is equal to $\frac{9}{4}$ times the intensity due to $S_{1}$ is: ( $n$ is a positive integer)

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$\frac{2 nλd}{\sqrt{d^{2} - n^{2} λ^{2}}}$

$\frac{d^{2} + n^{2} λ^{2}}{2 nλ}$

$\frac{nλd}{\sqrt{d^{2} - n^{2} λ^{2}}}$

$\frac{d^{2} - n^{2} λ^{2}}{2 nλ}$

answer is A.

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Detailed Solution

$l_{1} = 4 l_{0}, l_{2} = l_{0}, l_{P} = \frac{9}{4} l_{1} = 9 l_{0}$

$l_{P} = l_{1} + l_{2} + 2 \sqrt{l_{1} l_{2}} cosϕ$

$9 l_{0} = 4 l_{0} + l_{0} + 4 l_{0} cosϕ$

Or $Δx = nλ$
Or $\sqrt{x^{2} + d^{2}} - x = nλ (x = S_{1} P)$

Or $x^{2} + d^{2} = x^{2} + n^{2} λ^{2} + 2 xnλ$

$∴ x = \frac{d^{2} - n^{2} λ^{2}}{2 nλ}$

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