Question

While conducting Young's double slit experiment, a student replaced the two slits with a large opaque plate in the $x-y$ plane containing two small holes that act as two coherent point sources $(S_1,S_2)$ emitting light of wavelength $600\text{nm}$. The student mistakenly placed the screen parallel to the $x-z$ plane $($for $z>0)$ at a distance $D=3\text{m}$, from the mid-point of $S_1,S_2$, as shown schematically in the figure. The distance between the sources $d=0.6003\text{mm}$. The origin $O$ is at the intersection of the screen and the line joining $S_1,S_2$. Which of the following is(are) true of the intensity pattern on the screen?

• A. Straight bright and dark bands parallel to the $x$-axis.
• B. Semicircular bright and dark bands centered at point $O$.
• C. Hyperbolic bright and dark bands with foci symmetrically placed about $O$ in the $x$-direction.
• D. The region very close to the point $O$ will be dark.

Answer 1

Answer: (B), (D)

The correct options are
B Semicircular bright and dark bands centered at point $O$.
D The region very close to the point $O$ will be dark.


Fringes will be semicircular from YDSE theory.

The screen viewed from the front would look like

Here, every point in the semicircle will have the same path difference $\Delta x$.

Hence, semicircular bright and dark fringes will appear.
$\therefore$ options $\left(\text{C}\right)$ and $\left(\text{D}\right)$ are wrong.

Now to find the path difference at $O$ $\Delta x_o=d$

Let us assume that $n\lambda =d$

$n=\frac{d}{\lambda }=\frac{0.6003\text{mm}}{600\times {10}^{-9}\text{m}}=1000.5$

Hence, at $O$, $\Delta x=1000\lambda +\frac{\lambda }{2}$

Hence, it will be dark at $O$, which corresponds to minima. So we can say center point $O$ is minima and will be dark.

Therefore, the correct answer are $\left(\text{A}\right)$ and $\left(\text{B}\right)$.