Question

Two identical coherent sources placed on a diameter of a circle of radius $R$ at separation $x(\ll R)$ 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 $(x=5\lambda )$

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

Answer 1

The correct option is A $20$

Path difference at $P$ is
$\Delta x=2\left(\frac{x}{2}\cos \theta \right)=x\cos \theta$
For intensity to be maximum
$\Delta x=n\lambda$ $(n=0,1,\dots )$
$\Rightarrow x\cos \theta =n\lambda$
$\cos \theta =\frac{n\lambda }{x}$ or $\cos \theta \ngtr 1$
$\therefore \frac{n\lambda }{x}\ngtr 1$
$\therefore n\ngtr \frac{x}{\lambda }$
Substituting $x=5\lambda$
$n\ngtr 5$ or $n=1,2,3,4,5,\dots$
Therefore, in all four quadrants, there can be $2$ maximas. There are more maximas at $\theta =0^o$ and $\theta ={180}^o$.
But $n=5$ corresponds to $\theta ={90}^o$ and $\theta ={270}^o$ which are coming only twice. While we have multiplied it four times. Therefore, total number of maximas still $20$.