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 $(x=5 \lambda)$

• Correct answer
  A

  $20$
• B

  $22$
• C

  $24$
• D

  $26$

Solution

Path difference at $P$ is $\Delta x=2\left(\frac{x}{2} \cos \theta\right)=x \cos \theta$
For intensity to be maximum $\Delta x=n \lambda \quad(n=0,1, \ldots \ldots)$ $\Rightarrow \quad \begin{aligned} x \cos \theta & =n \lambda \\ \cos \theta & =\frac{n \lambda}{x} \text { or } \cos \theta \ngtr 1\end{aligned}$ $\begin{array}{ll}\therefore & \frac{n \lambda}{x} \ngtr 1 \\ \therefore & n \ngtr \frac{x}{\lambda}\end{array}$ $\begin{aligned} & \text { Substituting } x=5 \lambda \\ & \qquad n \ngtr 5 \text { or } n=1,2,3,4,5, \ldots \ldots .\end{aligned}$ 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.

Hence, the correct answer is option: 'A'.