Question

A helium-neon laser $(\mathrm{\lambda }=633\mathrm{nm})$ illuminates a diffraction grating. The distance between the two $\mathrm{m}=1$ bright fringes is $32\mathrm{cm}$ on a screen $2.0\mathrm{m}$ behind the grating. What is the spacing between slits of the grating$?$

Short answer

Distance between slits of the grating is$7.9\mathrm{\mu m}$.

Step-by-step solution

Formula for length between positions

The distance between the two$\mathrm{m}=1$spots will be,

$\mathrm{\Delta Y}=2{\mathrm{Ltan\theta }}_1$

$=2\mathrm{Ltan}\left({\sin }^{-1}\left(\frac{\mathrm{\lambda }}{\mathrm{d}}\right)\right)$

$\approx 2\mathrm{Ltan}\left(\frac{\mathrm{\lambda }}{\mathrm{d}}\right)$

$\mathrm{d}$is grating space.

$\mathrm{\lambda }$is wavelength.

${\mathrm{\theta }}_1$is diffraction angle

Calculation for spacing of grating

Small angle,

$\mathrm{x}<<1,\mathrm{x}\approx \sin \left(\mathrm{x}\right)\approx \tan \left(\mathrm{x}\right)$

Distance is,

$\tan \left(\frac{\mathrm{\lambda }}{\mathrm{d}}\right)=\frac{\mathrm{\Delta Y}}{2\mathrm{L}}$

$\Rightarrow \frac{\mathrm{\lambda }}{\mathrm{d}}={\tan }^{-1}\left(\frac{\mathrm{\Delta Y}}{2\mathrm{L}}\right)$

So,

$\mathrm{d}=\frac{\mathrm{\lambda }}{{\tan }^{-1}\left(\frac{\mathrm{\Delta Y}}{2\mathrm{L}}\right)}$

$=\frac{633·{10}^{-9}}{{\tan }^{-1}\left(\begin{array}{c}0.32\\ 2·2\end{array}\right)}$

$=7.9\mathrm{\mu m}$