Question

Two $50-\mu \mathrm{m}-$wide slits spaced $0.25mm$apart are illuminated by blue laser light with a wavelength of $450nm$. The interference pattern is observed on a screen$2.0m$ behind the slits. How many bright fringes are seen in the central maximum that spans the distance between the first missing order on one side and the first missing order on the other side?

Short answer

Number of bright fringes is$9$

Step-by-step solution

Introduction

The dazzling fringe occurs when the crest of one wave coincides with the crest of another. The dark fringe occurs when the trough of one wave coincides with the trough of another, resulting in dark fringes.

Find bright fringes 

The number of brilliant fringes between this order and the central maximum is $(m-1)$, for the missing order on one side of the central maximum. On the other hand, we have$(m-1)$bright fringes for the missing order $m$on the other side of the central maximum. As a result, the number of bright fringes seen between the first missing order on one side and the first missing order on the other side of the center maximum is $\left[2\right(m-1\left)\right]$.plus the central maximum itself, implying that the number is,

$n=2(m-1)+1$

$=2m-1$

Find bright fringes 

Our current objective is to locate$\left(m\right)$. The first missing order occurs when an interference maximum coincides with the first diffraction maximum, with $p$denoting the order of the minimum. As a result, when $p=1$, we can use the following relation to compute the missing order $m$.

$m_{\text{missing}}=p\frac{d}{a}p=1,2,3,\dots$

$$\begin{gathered} m=\frac{d}{a} \\ =\frac{0.25\times {10}^{-3}\mathrm{m}}{50\times {10}^{-6}\mathrm{m}} \\ =5 \end{gathered}$$

Therefore,

$$\begin{gathered} n=2m-1 \\ =10-1 \\ =9 \end{gathered}$$