Chapter 19.2, Problem 25PP

To determine

The number of slits per centimeter in the gratings.

Answer to Problem 25PP

  $1.60\times {10}^3\text{slits/cm}$

Explanation of Solution

Given:

The wavelength of the blue light, $\lambda =632\text{nm}$

The distance of the first bright band from the central bright band, $x=5.6\text{cm}$

The distance of the screen from the grating, $L=0.55\text{m}$

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.

And,

  $n=\frac{1}{d\left(\text{cm}\right)}$

Where, n is the number of slits per centimeter.

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,

  $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{x}{L}\right)\\ ={\tan }^{-1}\left(\frac{0.056}{0.55}\right)\\ =5.81°\end{array}$

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

  $\begin{array}{c}\lambda =d\sin \theta \\ 632\times {10}^{-7}\text{cm}=d\sin 5.81°\\ d=6.24\times {10}^{-4}\text{cm}\end{array}$

Then, the number of slits per centimeter on the grating will be,

  $\begin{array}{c}\text{number of slits per centimeter = }\frac{1}{6.24\times {10}^{-4}\text{cm}}\\ \approx 1.60\times {10}^3\text{slits/cm}\end{array}$

Conclusion:

Hence, the number of slits per centimeter on the grating is $1.60\times {10}^3$ .