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)

The total number of fringes produced between $P_1$ and $P_2$ in the first quadrant close to $3000$
At $P_2$ the order of the fringe will be maximum

Given $d=1.8\times {10}^{-3}m$ and $\lambda =6\times {10}^{-7}m$

Path difference at point P

$\Delta x=S_{1}P-S_{2}P=dsin\theta$ where $\theta$ angle is measured from vertical line as shown in figure.

For bright fridge $dsin\theta =m\lambda$ ------- 1

Point $P_1$ is the point of central maxima.

At point $P_2$, path difference $\left(\Delta x\right)=d$

If $P_2$ is the point of the fringe, then

$d=m\lambda ⟹m=\frac{d}{\lambda }=3000$

On differentiating equation (1)

$dcos\theta \left(\Delta \theta \right)=\left(\Delta m\right)\lambda$= constant for consecutive bright fringe.

$cos\theta \downarrow \therefore \Delta \theta \uparrow$ as $\theta$ varies from $0$ to $\frac{\pi }{2}$

Hence A and C are correct answer.