Question

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

• A. $26$
• B. $24$
• C. $22$
• D. $20$

Answer 1

The correct option is D $20$


The path difference at $\text{Q}$ is

$\Delta x=S_{1}P=x\cos \theta (\because x<<R)$

For intensity to be maximum

$\Delta x=n\lambda$

$\Rightarrow x\cos \theta =n\lambda$

$\Rightarrow \cos \theta =\frac{n\lambda }{x}$

As, $\cos \theta \ngtr 1$

$\Rightarrow \frac{n\lambda }{x}\ngtr 1$

$\Rightarrow \frac{n\lambda }{5\lambda }\ngtr 1$

$\Rightarrow n\ngtr 5$

$\therefore n=1,2,3,4,5$ in one quadrant or for $0\le \theta \le {90}^{\circ }$.

In four quadrants, points of maximum intensity $=4\times 5=20$

Hence, option $\left(\text{D}\right)$ is correct.