Isaac Barry

2022-09-04

Diffraction gratings with 10,000 lines per centimeter are readily available. Suppose you have one, and you send a beam of white light through it to a screen 2.00 m away.Find the angles for the first-order diffraction of the shortest and longest wavelengths of visible light (380 and 760 nm).

### Answer & Explanation

Sanaa Hudson

Given information:
The number of lines on the grating,
The distance between the screen and slits,
The shortest wavelength,
The longest wavelength,
To find
the angles corresponding to first-order diffraction from these wavelengths.
The grating equation is given by
$d\mathrm{sin}\left(\theta \right)=n\lambda \phantom{\rule{0ex}{0ex}}\mathrm{sin}\left(\theta \right)=\frac{n\lambda }{d}\phantom{\rule{0ex}{0ex}}\theta ={\mathrm{sin}}^{-1}\frac{n\lambda }{d}$
where,
d=width of the slit
n=order of diffraction
$\lambda =$ wavelength of incident light
$\theta =$ angle of diffraction
The width of the slit is calculated as follows

Apply the above formula for the shortest wavelength
${\theta }_{\text{min}}={\mathrm{sin}}^{-1}\left(\frac{1×380×{10}^{-9}}{{10}^{-6}}\right)\phantom{\rule{0ex}{0ex}}={\mathrm{sin}}^{-1}\left(0.38\right)\phantom{\rule{0ex}{0ex}}={22.33}^{\circ }$
Now, apply the above formula for the longest wavelength
${\theta }_{\text{max}}={\mathrm{sin}}^{-1}\left(\frac{1×760×{10}^{-9}}{{10}^{-6}}\right)\phantom{\rule{0ex}{0ex}}={\mathrm{sin}}^{-1}\left(0.76\right)\phantom{\rule{0ex}{0ex}}={49.46}^{\circ }$
The angle corresponding to the shortest wavelength is, ${\theta }_{\text{min}}={22.33}^{\circ }$
The angle corresponding to the longest wavelength is, ${\theta }_{\text{max}}={49.46}^{\circ }$