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