91Ó°ÊÓ

Number of lines per mm of the grating = 200

The angle between maximum intensity point and central axis = $30°$

The angular distance $\mathit{θ}$ of the $\mathit{m}\mathit{t}\mathit{h}$ order diffraction pattern produced from a grating having line separation $\mathit{d}$ is

$\mathit{d}\mathit{s}\mathit{i}\mathit{n}\mathit{θ}\mathbf{=}\mathit{m}\mathit{λ}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$

Here, $\mathit{λ}$ is the wavelength of the incident light.

The line separation of the grating is

$$\begin{gathered} \text{d}=\frac{1}{200}\text{mm} \\ =0.005\text{mm}\left(\frac{1n\text{m}}{{10}^{-7}mm}\right) \\ =5000\text{nm} \end{gathered}$$

From equation (1),

$$\begin{gathered} \mathrm{λ}=\frac{5000\text{nm}×\sin 30°}{m} \\ =\frac{5000\text{nm}×\frac{1}{2}}{m} \\ =\frac{2500\text{nm}}{m} \end{gathered}$$

The wavelength of visible light ranges from 400 nm to 800 nm. The wavelength corresponding to the first few values of m are

$$\begin{gathered} {\mathrm{λ}}_1=\frac{2500\text{nm}}{1} \\ =2500\text{nm} \end{gathered}$$

$$\begin{gathered} {\mathrm{λ}}_3=\frac{2500\text{nm}}{3} \\ =833.33\text{nm} \end{gathered}$$

$$\begin{gathered} {\mathrm{λ}}_4=\frac{2500\text{nm}}{4} \\ =625\text{nm} \end{gathered}$$

$$\begin{gathered} {\mathrm{λ}}_5=\frac{2500\text{nm}}{5} \\ =500\text{nm} \end{gathered}$$

$$\begin{gathered} {\mathrm{λ}}_6=\frac{2500\text{nm}}{6} \\ =416.67\text{nm} \end{gathered}$$

$$\begin{gathered} {\mathrm{λ}}_7=\frac{2500\text{nm}}{7} \\ =357.14\text{nm} \end{gathered}$$

Thus the wavelengths in the visible spectrum are 416.67 nm, 500 nm and 625 nm.

416.67 nm corresponds to violet light, 500 nm corresponds to cyan, and 625 nm corresponds to orange light.