Question

Two coherent monochromatic point sources $S_1$ and $S_2$ of wavelength $\lambda =600nm$ are placed symmetrically on either side of the center of the circle as shown. The sources are separated by a distance $d=1.8mm$. 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?

• A. The total number of fringes produced between $P_1$ and $P_2$ in the first quadrant close to $3000$
• B. A dark spot will be formed at the point $P_2$
• C. At $P_2$ the order of the fringe will be maximum
• D. The angular separation between two consecutive bright spots decreases as we move from $P_1$ to $P_2$ along the first quadrant

Answer 1

Answer: (A), (C)

At $P_2$ the order of the fringe will be maximum

Path difference at $P_2$ is $p=S_1{P}_2-S_2{P}_2=d=1.8mm=1.8\times {10}^{-3}m$
$=3000\times 600\times {10}^{-9}m$
$p=3000\lambda$

As the path ditterence is an integral multiple of $\lambda ,P_2$ should be a bright fringe with 300 th maxima. (B) is incorrect. Further at $P_1,$ path difference $=0.$ Therefore a bright fringe will be present at $P_1$ also. Therefore total number of fringes between $P_1$ and $P_2$ is $3000.\left(A\right)$ is a correct option. Obviously at $P_2$ the order of the fringe will be maximum. Thus (C) is a correct option.

Now path difference $p=d\cos \theta =n\lambda$
(for bright fringe)
$\therefore \cos \theta =\frac{n\lambda }{d}$
$\therefore -\sin \theta \Delta \theta =\left(\Delta n\right)\frac{\lambda }{d}$
or $\Delta \theta =-\frac{\Delta n\lambda }{d\sin \theta }$

As we move from $p_1$ to $p_2,\theta$ decreases and therefore $\Delta \theta$ increases. Therefore (D) is incorrect.