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Found in: Page 1115

### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# Show that the dispersion of a grating is ${\mathbit{D}}{\mathbf{=}}\frac{\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{\theta }}{\mathbf{\lambda }}$

It is proved that the dispersion of a grating is $\frac{\mathrm{tan}\theta }{\lambda }$

See the step by step solution

## Step 1: Definition and concept of diffraction from a grating

An optical element with a periodic structure that divides light into a number of beams that move in diverse directions is known as a diffraction grating.

The angular distance $\theta$of the ${m}_{th}$order diffraction pattern produced from a grating having line separation is

${\mathbit{d}}{\mathbit{s}}{\mathbit{i}}{\mathbit{n}}{\mathbit{\theta }}{\mathbf{=}}{\mathbit{m}}{\mathbit{\lambda }}$ …(i)

Here, $\lambda$ is the wavelength of the incident light.

## Step 2: Showing that dispersion of a grating  D=tanθλ

The dispersion is given by

$D=\frac{d\theta }{d\lambda }$

Differentiate equation (i) with respect to $\lambda$ to get

$d\mathrm{cos}\theta \frac{d\theta }{d\lambda }=m\phantom{\rule{0ex}{0ex}}\frac{d\theta }{d\lambda }=\frac{m}{d\mathrm{cos}\theta }$

Substitute form of $m$ from equation (i) to get