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

Answer: (A)

Path difference at $P$ is
$\Delta x=2\left(\frac{x}{2} \cos \theta\right)=x \cos \theta$

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