Question

A diffraction grating having $500lines/mm$diffracts visible light at ${30}^{\circ }$.What is the light's wavelength?

Short answer

The wavelength of sunshine is $\lambda =500nm.$

Step-by-step solution

Step: 1 Finding the grating spacing:

The constant value of grating is $500lines/mm$,the spacing by

$$\begin{gathered} d=\frac{1\mathrm{mm}}{500} \\ d=\underset{}{2\times {10}^{-6}\mathrm{m}}. \end{gathered}$$

Step: 2 Equating the equation:

In equation of grating,the light diffraction light wavelength by

$$\begin{gathered} d\sin \left({\theta }_m\right)=m\lambda \\ \Rightarrow \lambda =\frac{d\sin \left({\theta }_m\right)}{m}. \end{gathered}$$

Step: 3 Obtaining the wavelength value:

From expression of wavelength

$$\begin{gathered} \lambda =\frac{2\times {10}^{-6}\times \sin \left({30}^{\circ }\right)}{m} \\ \lambda =\frac{1000\mathrm{nm}}{m}. \end{gathered}$$

As we're seeing, there should only be one value of $m$that corresponds to a wavelength within the visible region and therfore the $m$will be $2$,The wavelength are$500nm$.