Question

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

Short answer

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

Step-by-step solution

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

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

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

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

$$\begin{gathered} d\cos \theta \frac{d\theta }{d\lambda }=m \\ \frac{d\theta }{d\lambda }=\frac{m}{d\cos \theta } \end{gathered}$$

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

$$\begin{gathered} \frac{d\theta }{d\lambda }=\frac{{\displaystyle \frac{d\sin \theta }{\lambda }}}{d\cos \theta } \\ \frac{d\theta }{d\lambda }=\frac{\tan \theta }{\lambda } \end{gathered}$$