Question 21: Wave Optics

Question

Young's double slit experiment is first done in air and then in a medium of refractive index $\mu$. If the 7 th dark fringe in the medium lies where the 4 th bright fringe is in air, then the value of $\mu$ is :

Accepted Answer

Let $y_{m}^{(	ext{bright, air})}$ be the position of the $m$th bright fringe in air, and $y_{n}^{(	ext{dark, med})}$ the position of the $n$th dark fringe in a medium of refractive index $\mu$. With slit separation $d$ and screen distance $L$,
In air (wavelength $\lambda$):
$$y_{m}^{(	ext{bright, air})} \;=\; m\,\frac{\lambda\,L}{d}.$$
In the medium the wavelength is $\lambda'=\lambda/\mu$, and the $n$th dark fringe sits at
$$y_{n}^{(	ext{dark, med})} \;=\;\Bigl(n+	frac12\Bigr)\,\frac{\lambda'}{d}\,L    \;=\;\Bigl(n+	frac12\Bigr)\,\frac{\lambda\,L}{\mu\,d}.$$
We are told the 7th dark fringe in the medium $(n=7)$ coincides with the 4th bright fringe in air $(m=4)$.  Equate their $y$-positions:
$ \Bigl(7+	frac12\Bigr)\,\frac{\lambda\,L}{\mu\,d} \;=\; 4\,\frac{\lambda\,L}{d} \quad\Longrightarrow\quad \frac{7.5}{\mu}=4 \quad\Longrightarrow\quad \mu=\frac{7.5}{4}=1.875. $
Answer: 1.875  (Option C)

Answer

1.654

Answer

1.389

Answer

1.875

Answer

1.768

Wave Optics

Young's double slit experiment is first done in air and then in a medium of refractive index \mu. If the 7 th dark fringe in the medium lies where the 4 th bright fringe is in air, then the value of \mu is :

Choose the correct answer: