Question

Two identical coherent sources placed on a diameter of a circle of radius $R$ at separation $x(<<R)$ symmetrically about the centre of the circle. The sources of points on the circle with maximum intensity is $(x=5\lambda )$.

• A. 20
• B. 22
• C. 24
• D. 26

Answer 1

Answer: (A)

From the figure, path difference $=S_{1}M=P$
$P=S_{1}M=x\cos \theta$
$(\because x<<R)$

$(S_{1}P$ and $S_{2}P$ are assumed approximately parallel)
For maximum intensity, $P=n\lambda$ (where, $n=0,1,2,3$ )
$\Rightarrow x\cos \theta =n\lambda$
$\Rightarrow \cos \theta =\frac{n\lambda }{x}$
$\Rightarrow \cos \theta =\frac{n\lambda }{5\lambda }$
$(\because x=5\lambda )$
$\Rightarrow \cos \theta =\frac{n}{5}$
We know, $-1\le \cos \theta \le 1$
$\Rightarrow -1\le \frac{n}{5}\le 1$
$\Rightarrow -5\le n\le 5$
Possible values of $n=\{0,\pm 1,\pm 2,\pm 3,\pm 4,\pm 5\}$
Let us analysis each value of $n$ for $\theta$ in range.
$\theta \in (0,2\pi )$
For $n=1,\cos \theta =\frac{1}{5}$

Here, negative value of $n$ means the path difference $\left(S_{1}P-S_{2}P\right)$ is negative, i.e., for those points $S_{1}P<S_{2}P$.
For $n=0,\pm 1,\pm 2,\pm 3,\pm 4$,
From the given graph of cosine function, it can be observed that in interval $\theta \in [0,2\pi ]$, for above values of $n$ there are in total 18 points, i.e., 2 points for $n=0,4$ points each for $n=\pm 1,\pm 2,\pm 3,\pm 4$.
For $n=+5,\cos \theta =+1$,
One value of $\theta$ i.e., $\theta =0^{\circ }$ is possible as for $\theta =2\pi$, the points will coincide.
For $n=-5,\cos \theta =-1$, i.e., $\theta =\pi$.
Thus, in total $20$ points of maxima's are possible in all $4$ quadrants.