1. Geometry of the setup

Let the circle have center $O$ and radius $R$. The two coherent point sources $S_1$ and $S_2$ are placed symmetrically about $O$ on a horizontal diameter, so we may take $S_1(-a,0),\qquad S_2(a,0),$ with source separation $d=2a=1.8\,\text{mm}.$

A point $P$ on the circumference at polar angle $\theta$ from the positive $x$-axis has coordinates $P(R\cos\theta, R\sin\theta).$

The labeled points in the first quadrant are naturally:

• $P_1$ near the positive $x$-axis: $\theta=0$
• $P_2$ near the positive $y$-axis: $\theta=\frac\pi2$

2. Path difference at a point on the circle

Using distance formula, $PS_1^2=(R\cos\theta+a)^2+(R\sin\theta)^2=R^2+a^2+2aR\cos\theta,$ $PS_2^2=(R\cos\theta-a)^2+(R\sin\theta)^2=R^2+a^2-2aR\cos\theta.$

Since $P$ lies on the circle and the two sources are symmetric, the path difference depends on $\theta$. For the standard circle-interference construction, the path difference is $\Delta = PS_1-PS_2 = d\cos\theta.$

So, $\boxed{\Delta = d\cos\theta}$ for points on the circumference.

3. Condition for bright and dark fringes

For bright spots, $\Delta = n\lambda,$ so $d\cos\theta = n\lambda.$ Hence the bright fringe angles satisfy $\cos\theta_n = \frac{n\lambda}{d}.$

For dark spots, $\Delta = \left(n+\frac12\right)\lambda,$ so $d\cos\theta = \left(n+\frac12\right)\lambda.$

Given: $d=1.8\times10^{-3}\,\text{m},\qquad \lambda=600\times10^{-9}\,\text{m}$ Therefore, $\frac{d}{\lambda}=\frac{1.8\times10^{-3}}{600\times10^{-9}}=3000.$

So the bright condition becomes $\cos\theta_n = \frac{n}{3000}.$

4. Check option B: order at $P_2$

At $P_2$, we have $\theta=\frac\pi2$, hence $\cos\frac\pi2=0$ so $\Delta = d\cos\frac\pi2 = 0.$ Thus the fringe order there is $n=0.$

Now in the first quadrant, $\cos\theta$ decreases from $1$ to $0$ as $\theta$ goes from $0$ to $\frac\pi2$. Therefore the order decreases from maximum to minimum.

So at $P_2$, the order is minimum, not maximum.

Hence option B is false.

5. Check option C: nature of point $P_2$

At $P_2$, $\Delta=0=0\cdot \lambda,$ which satisfies the bright condition. So $P_2$ is a bright spot, not a dark spot.

Hence option C is false.

6. Check option A: angular separation of consecutive bright spots

Bright spots satisfy $d\cos\theta=n\lambda.$ Differentiate with respect to $n$: $-d\sin\theta\, d\theta = \lambda\, dn.$ For consecutive bright spots, $dn=1$, so approximately $\Delta\theta \approx \frac{\lambda}{d\sin\theta}.$

As we move from $P_1$ ($\theta\approx 0$) to $P_2$ ($\theta\to \frac\pi2$), $\sin\theta$ increases from $0$ to $1$, so $\Delta\theta \propto \frac{1}{\sin\theta}$ decreases.

Hence option A is true.

7. Check option D: number of fringes between $P_1$ and $P_2$ in first quadrant

In the first quadrant, $0\le \theta\le \frac\pi2,$ so $0\le \cos\theta\le 1.$ Thus for bright spots, $0\le n\le \frac{d}{\lambda}=3000.$

This gives orders $n=0,1,2,\dots,3000,$ so the number of bright spots is $3001$ in that range.

Similarly, dark spots occur for $\cos\theta=\frac{n+1/2}{3000},$ which gives about $3000$ dark spots.

Thus the total fringe count (bright/dark alternations) between $P_1$ and $P_2$ is indeed of order $3000$ for each type, so the statement “close to 3000” is acceptable.

Hence option D is true.

8. Final evaluation of options

• A: True
• B: False
• C: False
• D: True

So the correct options are $\boxed{A,\ D}$

9. Comparison with stored answer

Stored correct answer: $B, D$

My derived answer is $A, D$.

I do not agree with the stored answer because at $P_2$ the path difference is zero, so it is a central bright fringe of order $0$, not maximum order.