Optics Ques 10
10 In a Young’s double slit experiment with slit separation 0.1 $\mathrm{mm}$, one observes a bright fringe at angle $\frac{1}{40}$ rad by using light of wavelength $\lambda_1$. When the light of the wavelength $\lambda_2$ $\lambda_2$ is used a bright fringe is seen at the same angle in the same set up. Given that $\lambda_1$ and $\lambda_2$ are in visible range ( $380 \mathrm{n}-\mathrm{m}$ to $740 \mathrm{n}-\mathrm{m}$ ), their values are
(2019 Main, 10 Jan I)
(a) $380 \mathrm{n}-\mathrm{m}, 525 \mathrm{n}-\mathrm{m}$
(b) $400 \mathrm{n}-\mathrm{m}, 500 \mathrm{n}-\mathrm{m}$
(c) $380 \mathrm{n}-\mathrm{m}, 500 \mathrm{n}-\mathrm{m}$
(d) $625 \mathrm{n}-\mathrm{m}, 500 \mathrm{n}-\mathrm{m}$
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Answer:
Correct Answer: 10.( d )
Solution:
$ \begin{aligned} & \qquad \Delta L=d \sin \theta \\ & \text { For small value of } \theta ; \sin \theta=\theta \\ & \text { So, path difference }=\Delta L=d \theta \end{aligned} $
For small value of $\theta ; \sin \theta=\theta$
For a bright fringe at same angular position ’ $\theta$ ‘, both of the rays from slits $S_1$ and $S_2$ are in phase.
Hence, path difference is an integral multiple of wavelength of light used.
i.e., $ \Delta L=n \lambda $ or $ d \theta=n \lambda \Rightarrow \lambda=\frac{d \theta}{n} $
Here, $\theta=\frac{1}{40} \mathrm{rad}, d=0.1 \mathrm{~mm}$
Hence, $\lambda=\frac{0.1}{40 n} \mathrm{~mm}=\frac{0.1 \times 10^{-3} \mathrm{~m}}{40 n}$
$ \begin{aligned} & =\frac{0.1 \times 10^{-3} \times 10^9}{40 n} \mathrm{n}-\mathrm{m} \\ \Rightarrow \quad \lambda & =\frac{2500}{n} \mathrm{n}-\mathrm{m} \end{aligned} $
So, with light of wavelength $\lambda_1$ we have
$ \lambda_1=\frac{2500}{n_1}(\mathrm{n}-\mathrm{m}) $
and with light of wavelength $\lambda_2$, we have
$ \lambda_2=\frac{2500}{n_2}(\mathrm{n}-\mathrm{m}) $
Now, choosing different integral values for $n_1$ and $n_2$, (i.e., $n_1, n_2=1,2,3 \ldots$ etc) we find that for
$ n_1=4, \lambda_1=\frac{2500}{4}=625 \mathrm{n}-\mathrm{m} $ and for $n_2=5$, $ \lambda_2=\frac{2500}{5}=500 \mathrm{n}-\mathrm{m} $
These values lie in given interval $500 \mathrm{n}-\mathrm{m}$ to $625 \mathrm{n}-\mathrm{m}$.
BETA
