Chapter 19, Problem 53A

To determine

The angle at which red and blue light have first-order bright lines.

Answer to Problem 53A

  $49.43°$ and $30.33°$ , respectively.

Explanation of Solution

Given:

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

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

Number of lines per centimeter, $n=12000\text{lines/cm}$

Formula used:

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

Where,

n is the number of lines in diffraction grating per centimeter and d is the distance between the slits.

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 distance between the slits will be,

  $\begin{array}{c}d=\frac{1}{n}\\ =\frac{1}{12000}\\ =8.33\times {10}^{-5}\text{cm}\end{array}$

Then, the angle at which the first-order bright line occurs for red light can be written as,

  $\begin{array}{c}{\theta }_r={\sin }^{-1}\left(\frac{{\lambda }_r}{d}\right)\\ ={\sin }^{-1}\left(\frac{632\times {10}^{-9}\text{m}}{8.33\times {10}^{-7}\text{m}}\right)\\ =49.43°\end{array}$

Similarly, the angle at which the first-order bright line occurs for blue light can be written as,

  $\begin{array}{c}{\theta }_b={\sin }^{-1}\left(\frac{{\lambda }_b}{d}\right)\\ ={\sin }^{-1}\left(\frac{421\times {10}^{-9}\text{m}}{8.33\times {10}^{-7}\text{m}}\right)\\ =30.33°\end{array}$

Conclusion:

Hence, the angle at which red and blue light have first-order bright lines is $49.43°$ and $30.33°$ , respectively.