91Ó°ÊÓ

Wavelength is defined as the distance between successive crests of a wave, especially points in a sound wave or electromagnetic wave.

(a)

Here initially there are 9 dark fringes between two ends.

Therefore we should have $L_R=\left(m+\frac{1}{2}\right)\frac{\mathrm{λ}}{2n}$

Here $n=1,m=9$ (since 9th dark fringe)

Wave length $\mathrm{λ}=620\text{nm}$

$$\begin{gathered} L_R=\left(9+\frac{1}{2}\right)\left(\frac{620\text{nm}}{2\left(1\right)}\right) \\ =2945 \end{gathered}$$

But in 15 s every bright fringe is changing to dark fringe.

So after 15 s the 9th dark fringe is changing to 9th bright fringe.

Therefore for bright fringe,

$$\begin{gathered} L_R^{\text{'}}=m\frac{\mathrm{λ}}{2n} \\ =\frac{\left(9\right)\left(620\text{nm}\right)}{2\left(1\right)} \\ =2790 \end{gathered}$$

Therefore changing in the width of the left edge is $\mathrm{Δ}{L}_R=L_R-L_R^{\text{'}}$

$$\begin{gathered} =2745\text{nm - 2790}\text{nm} \\ \text{= 155}\text{nm} \end{gathered}$$

This change takes place in $\mathrm{Δ}t=15\text{s}$

The rate of change of this spacing is

$$\begin{gathered} \frac{\mathrm{Δ}{L}_{\mathrm{τ}}}{\mathrm{Δ}t}\frac{155\text{nm}}{15\text{s}} \\ =10.33\text{nm/s} \end{gathered}$$

Hence, the rate of change of this spacing is $10.33nm/s$.

(b)

Here there are 5 dark fringes between left and right ends and each end have dark fringes. Therefore there are 7 dark fringes.

Therefore $m=6$

role="math" localid="1663148654843" $$\begin{gathered} L_R^{\text{'}}=\left(6\right)\frac{\mathrm{λ}}{2n} \\ =\frac{\left(620\text{nm}\right)}{\left(2\right)\left(1\right)} \\ =1860\text{nm} \end{gathered}$$

Change in the spacing is

$$\begin{gathered} L_R-L_R^{\text{'}}=2945\text{nm - 1860}\text{nm} \\ \text{= 1085}\text{nm} \end{gathered}$$

Hence, the change in the spacing is $1085nm$.