Question

Figure shows two coherent microwave source $S_1$ and $S_2$ emitting waves of wavelength $\lambda$ and separated by a distance $3\lambda$. For $\lambda <<D$ and $y\ne 0$,the minimum value of $y$ for point $P$ to be an intensity maximum is $\frac{D\sqrt{m}}{n}$ Determine the value of $m+n,$ if $m$ and $n$ are coprime numbers.

Answer 1

We know that
$\Delta x=S_{1}P-S_{2}P$
$\Rightarrow S_{1}Q\Rightarrow 3\lambda \cos \theta$
$\Delta x=\lambda$
$\therefore 3\lambda \cos \theta =\lambda$
For the same intensity point
$\cos \theta =\frac{1}{3}$
From $\Delta P{S}_{1}O$
$\tan \theta =\frac{y}{D}$

$y=D\times 2\sqrt{2}$

So, this is not correct.

Similarly,
$3\lambda \cos \theta =2\lambda$

From $\Delta P{S}_{1}O$ , $y=\frac{D\sqrt{5}}{2}$

Given,
$y=\frac{D\sqrt{m}}{n}$

$y=\frac{D\sqrt{5}}{2}$

$m=5$

$n=2$

$m+n=7$