Chapter 24, Problem 72GP

To determine

The number of lines/cm is on the grating.

Answer to Problem 72GP

  $938\text{lines/cm}$

Explanation of Solution

Given data:

Distance, $y_1=3.32\text{cm}$

Distance, $y_2=3.71\text{cm}$

Wavelength, $\lambda =589\times {10}^{-9}\text{m}$

Screen length, $\text{L}=60.0\text{cm}$

Formula used:

  $d\sin \theta =m\lambda$ and $\text{tanθ}=\frac{\text{y}}{\text{L}}$

Where,

  $d$ is width of the slit.

  $\lambda$ is wavelength.

  $y$ is distance.

  $\theta$ is an angle.

  $\text{L}$ is the screen length.

Calculation:

The angles for the first-order form the given distances, $\text{tanθ}=\frac{\text{y}}{\text{L}}$

  ${\text{tanθ}}_1=\frac{3.32\text{cm}}{60.0cm}=0.0553$

  $\therefore {\text{θ}}_1=3.17°$

  ${\text{tanθ}}_2=\frac{3.12\text{cm}}{60.0cm}=0.0618$

  $\therefore {\text{θ}}_2=3.54°$

Now, we calculate the separation of lines form, $d\sin \theta =m\lambda$ , we get,

  $d\sin 3.17°=1\left(589\times {10}^{-9}\text{m}\right)$

  $d=1.066\times {10}^{-5}\text{m}=1.066\times {10}^{-3}\text{cm}$

For the second wavelength,

  $\left(1.066\times {10}^{-3}\text{cm}\right)\sin 3.17°=1\times {\lambda }_2$

  ${\lambda }_2=6.58\times {10}^{-7}\text{m}=658\text{nm}$

The number of the lines/cm:

  $\frac{1}{\text{d}}=\frac{1}{1.066\times {10}^{-3}\text{cm}}$

  $\therefore \text{d}=\text{938}\text{lines/cm}$

Conclusion: The number of the lines is $\text{938}\text{lines/cm}$ .