Dispersive power of plane diffraction grating and its expression

Dispersive power of plane diffraction grating:

The dispersive power of a diffraction grating is defined as:

The rate of change of the angle of diffraction with the change in the wavelength of light are called dispersive power of plane grating.

If the wavelenght changes from $\lambda$ to $\lambda +d\lambda$ and respective change in the angle of diffraction be from $\theta$ to $\theta+d\theta$ then the ratio $\left(\frac{d\theta}{d\lambda} \right)$

Expression of Dispersive power of a plane diffraction grating:

The grating equation for a plane transmission grating for normal incidence is given by

$(e+d)sin\theta=n\lambda \qquad(1)$

Where$(e+d)$ - Grating Element$\qquad \:\: \theta$ - Diffraction angle for spectrum of $n^{th}$ order

Differentiating equation $(1)$ with respect to $\lambda$, we have

$(e+d)cos\theta \left( \frac{d\theta}{d\lambda} \right)=n$

$\frac{d\theta}{d\lambda}=\frac{n}{(e+d)cos\theta}$

$\frac{d\theta}{d\lambda}=\frac{n}{(e+d)\sqrt{1-sin^{2}\theta}} \qquad(2)$

Now substitute the value of $sin\theta$ from equation$(1)$ in equation$(2)$

$\frac{d\theta}{d\lambda}=\frac{n}{(e+d)\sqrt{1- \frac{n^{2}\lambda^{2}}{(e+d)^{2}}}} $

$\frac{d\theta}{d\lambda}=\frac{1}{\sqrt{\left(\frac{e+d}{n} \right)^{2}}- \lambda^{2}}$

Here $d\theta$- Angular separation between two lines

The above equation gives the following conclusions:

The dispersive power is directly proportional to the order of spectrum$(n)$

The dispersive power is inversely proportional to the grating element $(e+d)$.

The dispersive power is inversely proportional to the $cos\theta$ i.e Larger value of $\theta$, higher is the dispersive power.