Q11E - University Physics with Modern Physics

Question

Red light of wavelength 633 nm from a helium-neon laser passes through a slit 0.350 mm wide. The diffraction pattern is observed on a screen 3.00 m away. Define the width of a bright fringe as the distance between the minima on either side. 

(a) What is the width of the central bright fringe? 

(b) What is the width of the first bright fringe on either side of the central one?

Step-by-Step Solution

Answer

1. The width of the central bright fringe is 10.84 mm.
2. The width of the first bright fringe on either side of the central one is 5.42 mm.

Step 1: Given.

$\lambda =633nm=633\times {10}^{-9}m,a=0.350mm=0.350\times {10}^{-3}m,R=3.00m$

Step 2: Calculate the width of the first bright fringe on either side of the central one

We are given with the sit separation d = 0.350 mm, the distance from slits to screen D =3 m, and the wavelength $\lambda =633nm$. So for the width of the first bright fringe on either side of the central one

$d \sin \theta =m\lambda$ (1)

For smaller $\theta , \sin \theta \approx \theta$ the equation is

                                          $\theta =\frac{y}{D}$

By substituting the value of in equation (1), we get

$$\begin{gathered} d\frac{y}{D}=m\lambda \\ y=\frac{\left(m\right)\left(\lambda \right)\left(D\right)}{d} \\ =\frac{\left(\mathrm{\lambda }\right)\left(\mathrm{D}\right)}{\mathrm{d}} \mathrm{m}=1\mathrm{in} \mathrm{case} \mathrm{of} \mathrm{central} \mathrm{bright} \mathrm{fringe} \\ =\frac{(633\times {10}^{-9})\left(3\right)}{0.350\times {10}^{-3}} \\ =5.42\times {10}^{-3}m \\ =5.42mm \end{gathered}$$

Step 3: Calculate the width of the central bright fringe.

The width of the central bright fringe will be 2y of the first bright fringe on either side of the central one. So

                                                         $2y=2\times 5.42mm=10.84mm$