91Ó°ÊÓ

A set of a large number of slits is called a diffraction grating to resolve the incident light into its component wavelengths. The diffractions at angles $\mathit{θ}$ for N slits are given by

$\mathit{d}\mathit{s}\mathit{i}\mathit{n}\mathit{θ}\mathbf{=}\mathit{m}\mathit{λ}\mathbf{}\mathbf{}\mathbf{}\mathbf{for}\mathbf{}\mathit{m}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{}\mathbf{1}\mathbf{,}\mathbf{}\mathbf{2}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}$(maxima)

Where d is the width of the grating element, which is equal to $\frac{\mathbf{w}\mathbf{i}\mathbf{d}\mathbf{t}\mathbf{h}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{g}\mathbf{r}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{n}\mathbf{g}}{\mathbf{N}\mathbf{u}\mathbf{m}\mathbf{b}\mathbf{e}\mathbf{r}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{s}\mathbf{l}\mathbf{i}\mathbf{t}\mathbf{s}}$.

And the resolving power R of two observed wavelengths is

$\mathit{R}\mathbf{=}\frac{{\mathbf{λ}}_{\mathbf{a}\mathbf{v}\mathbf{g}}}{\mathbf{Δ}\mathbf{λ}}\mathbf{=}\mathit{N}\mathit{m}$

Where${\mathit{λ}}_{\mathbf{a}\mathbf{v}\mathbf{g}}$ is the average of the two wavelengths, and$\mathit{Δ}\mathit{λ}$is wavelength width.

Two wavelengths that are resolved with diffraction grating are 590.159 nm and 590.220 nm. The average of these is

$$\begin{gathered} {\mathrm{λ}}_{avg}=\frac{{\mathrm{λ}}_1+{\mathrm{λ}}_2}{2} \\ =\frac{590.159nm+590.220nm}{2} \\ =590.190nm \end{gathered}$$

And the wavelength width will be

$$\begin{gathered} \mathrm{Δ}\mathrm{λ}={\mathrm{λ}}_2-{\mathrm{λ}}_1 \\ =590.220nm-590.159nm \\ =0.061nm \end{gathered}$$

Inserting these two terms in the resolving power equation to determine the number of slits

$$\begin{gathered} \frac{{\mathrm{λ}}_{avg}}{\mathrm{Δ}\mathrm{λ}}=Nm \\ N=\frac{{\mathrm{λ}}_{avg}}{m\mathrm{Δ}\mathrm{λ}} \\ N=\frac{590.190nm}{2\left(0.061nm\right)} \\ N≈4838 \end{gathered}$$

Hence the minimum number of rulings required is 4838.