Chapter 19.2, Problem 22PP

To determine

The spacing between the slits in the grating.

Answer to Problem 22PP

  $9.35\times {10}^{-7}\text{m}$

Explanation of Solution

Given:

The wavelength of the blue light, $\lambda =434\text{}\text{nm}$

The spacing of the resulting lines on a screen, $x=0.55\text{m}$

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.

Calculation:

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

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

Then, using the given values in above,

  $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{x}{L}\right)\\ ={\tan }^{-1}\left(\frac{0.55}{1.05}\right)\\ =27.65°\end{array}$

Then, using the mentioned formula and the given values, the spacing between the slits in the grating will be,

  $\begin{array}{c}\lambda =d\sin \theta \\ 434\times {10}^{-9}=d\sin 27.65°\\ d=9.35\times {10}^{-7}\text{m}\end{array}$

Conclusion:

Hence, the spacing between the slits in the grating is $9.35\times {10}^{-7}\text{m}$ .