Question

Two coherent sources $S_1$ and $S_2$ are emitting light of wavelength $5000\dot{\text{A}}$ are placed at $0.1\text{mm}$ apart as shown in the figure. A detector is moved along a line perpendicular to $S_1{S}_2$ and passing through $S_1$. The position of the farthest maxima from $S_1$ is approximately at a distance of

• A. $1.4\text{cm}$
• B. $2\text{cm}$
• C. $1.2\text{cm}$
• D. $1\text{cm}$

Answer 1

The correct option is D $1\text{cm}$

At the position of the maxima, the path difference is

$\Delta x=n\lambda$

$\Rightarrow S_{2}P-S_{1}P=n\lambda$

$\Rightarrow \sqrt{d^2+D^2}-D=n\lambda$

$\Rightarrow d^2+D^2=(D+n\lambda {)}^2$

$\Rightarrow d^2+D^2=D^2+n^2{\lambda }^2+2n\lambda D$

$\Rightarrow 2n\lambda D=d^2\left[{\lambda }^2\text{being very small is neglected}\right]$

$\therefore D=\frac{d^2}{2n\lambda }=\frac{{10}^{-8}}{2\times 5000\times {10}^{-10}}=\frac{{10}^{-2}}{n}$

$\Rightarrow D=\frac{1\text{cm}}{n}$

$\therefore D=1\text{cm},0.5\text{cm},0.25\text{cm},0.125\text{cm}...$

So, the position of farthest maxima from $S_1$ will be at $D=1\text{cm}.$

Hence, option $\left(\text{B}\right)$ is correct.