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<
$\left(S_{1} P\right.$ 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, $-1 \leq \cos \theta \leq 1$
$\Rightarrow -1 \leq \frac{n}{5} \leq 1$
$\Rightarrow -5 \leq n \leq 5$
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.