Question

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?

Short answer

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

Step-by-step solution

Identification of given data

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, $m=3$

Concept/Significance of diffraction

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.

Determination of the smallest wavelength difference this grating can resolve in the third order

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 $5.95\times {10}^{-11}\text{m}$.