Question

FIGURE P33.49 shows the interference pattern on a screen $1.0m$behind a diffraction grating. The wavelength of the light is $620nm$. How many lines per millimeter does the grating have?

Short answer

The grating has$n=645nm$of lines per millimetre.

Step-by-step solution

Step: 1 Derivating:

The bright fringes diffraction grating as

$\begin{array}{l}d\sin \left({\theta }_m\right)=m\lambda \\ d=\frac{m\lambda }{\sin \left({\theta }_m\right)}\\ d=\frac{\lambda }{\sin \left({\theta }_1\right)}.\end{array}$

Step: 2 Finding the angle:

The place of bright fringes as

$\begin{array}{l}\tan \left({\theta }_1\right)=\frac{y_1}{L}\\ {\theta }_1={\tan }^{-1}\left(\frac{y_1}{L}\right)\\ {\theta }_1={\tan }^{-1}\left(\frac{43.6\mathrm{cm}}{100\mathrm{cm}}\right)\\ {\theta }_1={23.56}^{\circ }.\end{array}$

Step: 3 Calculating grating value:

The spacing and grating value as

$$\begin{gathered} d=\frac{620\times {10}^{-9}\mathrm{m}}{\sin \left({23.56}^{\circ }\right)} \\ d=1.55\times {10}^{-6}\mathrm{m} \\ \mathrm{d}=1.55\times {10}^{-3}\mathrm{mm}. \end{gathered}$$

The number of lines is

$$\begin{gathered} n=\frac{1}{1.55\times {10}^{-3}\mathrm{mm}} \\ n=645. \end{gathered}$$