Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson
ISBN 10: 0321740904
ISBN 13: 978-0-32174-090-8

Chapter 22 - Wave Optics - Exercises and Problems - Page 650: 36

Answer

$1.7\;\rm cm$

Work Step by Step

First of all, we need to sketch this problem, as we see below. We can see that there are two path length differences now due to the tilting of the laser beam. $$\Delta r_1=d\sin\theta_1$$ and $$\Delta r_2=d\sin\theta_2$$ And since we need the diffracted rays to interfere constructively to create the central maximum, then the total path length difference between the two rays is then given by an integer number of light wavelengths. $$\Delta r_1+\Delta r_2=m\lambda$$ Thus, $$d\sin\theta_1+d\sin\theta_2=m\lambda$$ And since $m=0$ at central maximum, $$d\sin\theta_1+d\sin\theta_2=m\lambda=0$$ $$\sin\theta_1=-\sin\theta_2\tag 1$$ Now we need to find $y_f$ the final height of the central maximum which is given by $$y=L\tan\theta_1$$ where $\theta_1=\sin^{-1}\left[ -\sin\theta_2 \right]$, from (1) Thus, $$y=L\tan\left(\sin^{-1}\left[ -\sin\theta_2 \right]\right)$$ Plugging the known; $$y=(1)\tan\left(\sin^{-1}\left[ -\sin1^\circ \right]\right)=\bf -0.017\;\rm m$$ thus the central maximum will be shifted by a distance of $$y=\color{red}{\bf 1.7}\;\rm cm$$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.