Question

A diffraction grating has slit spacing $d$. Fringes are viewed on a screen at distance $L$. Find an expression for the wavelength of light that produces a first-order fringe on the viewing screen at distance$L$from the center of the screen.

Short answer

The wavelength of light is$\lambda =d\frac{\sqrt{2}}{2}.$

Step-by-step solution

Step: 1 Diffraction grating:

A light beam is created by scratching a flat piece of transparent material with multiple parallel scratches. The material may be scratched with a great number of scratches per centimetre.

Step: 2 Angle:

The $m^{th}$fringe as

$\begin{array}{l}{Y}_m=L\tan \left({\theta }_m\right),\\ \theta ={\tan }^{-1}\left(\frac{Y}{L}\right)\end{array}$

Step: 3 Obtaining wavelength:

The wavelength from grating equation as

$\begin{array}{l}d\sin \left({\theta }_m\right)=m\lambda \\ \lambda =d\sin \left({\theta }_1\right)\\ \lambda =d\frac{\sqrt{2}}{2}.\end{array}$