Question

A diffraction grating has ${}^{\mathbf{2000}}$ lines per centimeter. At what angle will the first-order maximum be for ${}^{\mathbf{520}\mathbf{}\mathbf{nm}}$ -wavelength green light?

Short answer

The first order maximum is obtained as ${}^{\mathbf{5}\mathbf{.}{\mathbf{97}}^{\mathbf{0}}}$.

Step-by-step solution

Given data

Grating lines on the diffraction grating are 2000 liner per cm, which means that the distance between the two slits is$\mathrm{d}=\left(\frac{1\mathrm{cm}}{2000}\right)\left(\frac{1\mathrm{m}}{100\mathrm{cm}}\right)=5.0\times {10}^{-6}\mathrm{m}$

The wavelength of the green light is$\mathrm{\lambda }=520\mathrm{nm}\left(\frac{{10}^{-9}\mathrm{m}}{1\mathrm{nm}}\right)=5.20\times {10}^{-7}\mathrm{m}$

The order of maximum is 1.

Evaluating the first-order maximum

In order to calculate the angle, we are supposed to firstly find the distance between slits in the grating, for the value of ${}^{\mathrm{d}}$, and then solve the equation of the constructive interference,

$\mathrm{d}\mathrm{sin\theta }=\mathrm{m\lambda }$

The value here is for ${}^{\mathrm{\theta }}$when ${}^{\mathrm{m}=1}$.

$$\begin{gathered} \mathrm{\theta }={\sin }^{-1}\left(\frac{\mathrm{\lambda }}{\mathrm{d}}\right) \\ ={\sin }^{-1}\left(\frac{5.2\times {10}^{-7}\mathrm{m}}{5.0\times {10}^{-6}\mathrm{m}}\right) \\ ={\sin }^{-1}(0.104) \\ =5.{97}^0 \end{gathered}$$

Therefore, the first-order maximum is ${}^{5.{97}^0}$.