Chapter 27, Problem 10TP

To determine

The nth -order differection angle for the shortest wavelength and change in two angles if the distance between the screen and the grating is doubled.

Answer to Problem 10TP

The nth -order differection angle for the shortest wavelength is $23.578°$ and change in angle for the longest wavelength and shortest wavelength if the distance between the screen and the grating is doubled is $29.552°$ and $12.042°$ respectively.

Explanation of Solution

Given:

The nth -order diffraction angle for the longest wavelength is, ${\theta }_{l1}=53.13°$.

The longest wavelength is, ${\lambda }_l=760nm$.

The shortest wavelength is, ${\lambda }_s=380nm$.

Formula used:

The wavelength of constructive interference is given by,

  $\lambda =\frac{d\sin \theta }{m}$

The new distance between slits in given by,

  $d^{\prime }=2d$

The change in angle is given by,

  ${\theta }_d=\left|{\theta }_1-{\theta }_2\right|$

Calculation:

The distance between slits is calculated as,

  $\begin{array}{c}d=\frac{m\lambda }{\sin \theta }\\ =\frac{n\cdot 760nm}{\sin 53.13}\\ =\left(760nm\times 1.25\right)n\\ =950nnm\end{array}$

The nth -order differection angle for the shortest wavelength is calculated as,

  $\begin{array}{c}{\theta }_{s1}={\sin }^{-1}\left(\frac{m\lambda }{d}\right)\\ ={\sin }^{-1}\left(\frac{n\cdot 380nm}{950nnm}\right)\\ =23.578°\end{array}$

The new distance between the slits is calculated as,

  $\begin{array}{c}{d}^{\prime }=2d\\ =2\times 950nnm\\ =1900nnm\end{array}$

The nth -order differection angle for the longest wavelength when $d$ is doubled is calculated as,

  $\begin{array}{c}{\theta }_{l2}={\sin }^{-1}\left(\frac{m\lambda }{d^{\prime }}\right)\\ ={\sin }^{-1}\left(\frac{n\cdot 760nm}{1900nnm}\right)\\ =23.578°\end{array}$

The nth -order differection angle for the shortest wavelength when $d$ is doubled is calculated as,

  $\begin{array}{c}{\theta }_{s2}={\sin }^{-1}\left(\frac{m\lambda }{d^{\prime }}\right)\\ ={\sin }^{-1}\left(\frac{n\cdot 380nm}{1900nnm}\right)\\ =11.536°\end{array}$

The change in angle for longest wavelength is calculated as,

  $\begin{array}{c}{\theta }_l=\left|{\theta }_{l1}-{\theta }_{l2}\right|\\ =\left|53.13°-23.578°\right|\\ =29.552°\end{array}$

The change in angle for shortest wavelength is calculated as,

  $\begin{array}{c}{\theta }_s=\left|{\theta }_{s1}-{\theta }_{s2}\right|\\ =\left|23.578°-11.536°\right|\\ =12.042°\end{array}$

Conclusion:

The nth -order differection angle for the shortest wavelength is $23.578°$ and change in angle for the longest wavelength and shortest wavelength if the distance between the screen and the grating is doubled is $29.552°$ and $12.042°$ respectively.