A beam of light with a narrow wavelength range centered on 450 nm is incident perpendicularly on a diffraction grating with a width of 1.80 cm and a line density of 1400 lines/cm across that width. For this light, what is the smallest wavelength difference this grating can resolve in the third order?

The given data can be listed below,

• The wavelength of the light beam is, $\lambda =450\text{nm}$
• The length of diffraction grating is,$d=1.80\text{cm}$
• The line density across the width is,$L=1400\text{line/cm}$
• The order of diffraction is, $m=3$

When light passes through a slit, it will diffract by the slit's edge and deviate from its intended direction, which is in a straight line. This occurrence is related to the object's wavelike nature. Therefore, the diffraction is a deviation of the wave from its linear course.

The number of diffraction lines is given by,

$N=Ld$

Here, Lis the line density across the width, and d is the length of the diffraction grating.

Substitute all the values above,

$$\begin{gathered} N=1400\text{lines/cm}\times 1.80\text{cm} \\ =2520\text{lines} \end{gathered}$$

The change in path difference is given by,

$\Delta \lambda =\frac{\lambda }{Nm}$

Here,$\lambda$ is the wavelength of the light beam, N is the number of diffraction lines, m is the mass

Substitute all the values in the above,

$$\begin{gathered} \Delta \lambda =\frac{\left(450\times {10}^{-9}\text{m}\right)}{2520\text{m}\left(3\right)} \\ =5.95\times {10}^{-11}\text{m} \end{gathered}$$

Thus, the smallest change in wavelength of the grating is $5.95\times {10}^{-11}\text{m}$.