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