Chapter 19.2, Problem 23PP

To determine

The separation of the lines in the diffraction pattern.

Answer to Problem 23PP

  $0.58\text{m}$

Explanation of Solution

Given:

The wavelength of the violet light, $\lambda =421\text{}\text{nm}$

The distance between the slits on the diffraction grating, $d=8.60\times {10}^{-7}\text{m}$

The distance of the screen from the grating, $L=1.05\text{m}$

Formula used:

Wavelength of light can be obtained by:

  $\lambda =d\sin \theta$

Where,

  $\theta$ is the angle at which the first order bright line occurs

d is the distance between the slits.

The angle at which the first-order bright line occurs can also be written as,

  $\theta ={\tan }^{-1}\left(\frac{x}{L}\right)$

Where,

x is the separation of the lines in the diffraction pattern

L is the distance between the grating and screen.

Calculation:

Now, the angle at which the first-order bright line occurs can be written as,

  $\lambda =d\sin \theta$

Then, using the given values in above,

  $\begin{array}{c}\theta ={\sin }^{-1}\left(\frac{\lambda }{d}\right)\\ ={\sin }^{-1}\left(\frac{421\times {10}^{-9}}{8.60\times {10}^{-7}}\right)\\ =29.27°\end{array}$

Then, using the mentioned formula and the given values, the separation of the lines in the diffraction pattern will be,

  $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{x}{L}\right)\\ \tan \left(29.27°\right)=\frac{x}{1.05}\\ x=0.58\text{m}\end{array}$

Conclusion:

Hence, the separation of the lines in the diffraction pattern is $0.58\text{m}$ .