Two coherent monochromatic point sources S₁ and S₂​ of wavelength $\lambda =600 nm$ are placed symmetrically on either side of the center of the circle as shown. The sources are separated by a distance d=1.8 mm. This arrangement produces interference fringes visible as alternate bright and dark spots on the circumference of the circle. The angular separation between two consecutive bright spots is $\Delta \theta$. Which of the following options is/are correct? 

From the figure, we observe that the path difference between the waves arriving at the circle from sources S₁ and S₂ is $x=d\sin \theta$

At P₁, $\theta =0°$

$\Rightarrow x=0$

At P₂, $\theta =90°$

$\Rightarrow x=d$

Since for maxima, we have $d\sin \theta =n\lambda$

$\Rightarrow \sin \theta =\frac{n\lambda }{d}$

Since at P₂, $\theta =90°$

$$\begin{gathered} \Rightarrow \frac{n\lambda }{d}=1 \\ \Rightarrow n=\frac{d}{\lambda }=\frac{1.8\times {10}^{-3}}{600\times {10}^{-9}}=3000 \end{gathered}$$

So total number of fringes produced between P₁ and P₂ in first quadrant is close to 3000.

Since, $x=d\sin \theta$

$\Rightarrow ∆x=∆\left(d\sin \theta \right)$

Since for maxima, we have $x=\left(2n\right)\frac{\lambda }{2}=n\lambda$

$$\begin{gathered} \Rightarrow ∆\left(n\lambda \right)=d∆\left(\sin \theta \right) \\ \Rightarrow \lambda ∆n=\left(d\cos \theta \right)∆\theta \end{gathered}$$

Now, between any two consecutive maxima, we have

$$\begin{gathered} \left|∆n\right|=1 \\ \Rightarrow \lambda =\left(d\cos \theta \right)∆\theta \\ \Rightarrow ∆\theta =\frac{\lambda }{d\cos \theta } \end{gathered}$$

Now, as we move from P₁ to P₂, $\theta$ increases, $\cos \theta$ decreases and hence $∆\theta$ increases.