Question

For a grating, how many lines per millimeter would be required for the first-order diffraction line for $𝛌=400\mathrm{nm}$to be observed at a reflection angle of $7°$when the angle of incidence is$45°$?

Short answer

The number of lines per millimeter for grating for first-order diffraction is$2072\mathrm{lines}/\mathrm{mm}$.

Step-by-step solution

Given information

$$\begin{gathered} 𝛌=400nm \\ r=7° \\ i=45° \end{gathered}$$

Determine the number of lines per millimeter for grating for first-order diffraction.

Solution

Formula is,

$$\begin{gathered} n𝛌=d(\sin i+\sin r) \\ d=\frac{n𝛌}{(\sin i+\sin r)} \\ d=\frac{1\times 400\times {10}^{-9}}{(\sin 45°+\sin 7°)} \\ d=\frac{400\times {10}^{-9}}{0.8289} \\ d=4.825\times {10}^{-7} \end{gathered}$$

Lines per millimeter is,

$$\begin{gathered} \mathrm{Number}\mathrm{of}\mathrm{Line}=\frac{1}{4.825\times {10}^{-7}}\times {10}^{-3}\mathrm{m} \\ =2072\mathrm{lines}/\mathrm{mm} \end{gathered}$$