College Physics (7th Edition)

Published by Pearson
ISBN 10: 0-32160-183-1
ISBN 13: 978-0-32160-183-4

Chapter 24 - Physical Optics: The Wave of Nature of Light - Learning Path Questions and Exercises - Exercises - Page 841: 12


a). Location of minima = $y_{n}=(n+\frac{1}{2})\frac{vL}{d}$, n=1,2,3, ..... Distance between adjacent minima is $\frac{vL}{d}$ b). Path length difference = $\frac{5}{2}\frac{v}{d}$

Work Step by Step

a). To derive a relation which gives location of minima in Young's double slit experiment. v=wavelength $\Delta L=\frac{mv}{2}$, m=1,3,5, .... $dsin\theta=\Delta L=\frac{mv}{2}$, m=1,3,5 .... Again, $dsin\theta=nv$, n=0,1,2,3 ..... Comparing above two , $m=(2n+1)$ or, $dsin\theta=(n+\frac{1}{2})v$, n=0,1,2,3,.... Condition for minima Taking the approximation for small $\theta$, $tan\theta=sin\theta=\frac{y}{L}$ So, $d\frac{y_{n}}{L}=(n+\frac{1}{2})v$, n=1,2,3, .... or, $y_{n}=(n+\frac{1}{2})\frac{vL}{d}$, n=1,2,3, ..... This gives location of minima. The distance between adjacent minima = $y_{n+1}-y_{n}=(n+1+\frac{1}{2})\frac{vL}{d}-(n+\frac{1}{2})\frac{vL}{d}=\frac{vL}{d}$ b). For third order minimum we need to find path length difference between that location and the two slits. From a, $y_{n}=(n+\frac{1}{2})\frac{vL}{d}$ or, $\frac{y_{n}}{L}=(n+\frac{1}{2})\frac{v}{d}$ For third minimum, n=2, $\frac{y_{2}}{L}=(2+\frac{1}{2})\frac{v}{d}=\frac{5}{2}\frac{v}{d}$
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