Question

A laser beam illuminates a single, narrow slit, and the diffraction pattern is observed on a screen behind the slit. The first secondary maximum is $26mm$from the center of the diffraction pattern. How far is the first minimum from the center of the diffraction pattern?

Short answer

Minimum from the center of the diffraction pattern,$y_1=18.2\mathrm{mm}$

Step-by-step solution

Introduction

A laser beam is a single-wavelength stream of concentrated, coherent light.

Find maximum distance

Setting the derivative of the following equation to zero and solving the resultant equation will reveal the position of the first and second secondary maxima.

$I_{\text{slit}}=I_0{\left[\frac{\sin (\pi ay/\lambda L)}{\pi ay/\lambda L}\right]}^2$

where the first maximum's position is judged to be

$y_{\max ,1}=1.43\left(\frac{\lambda L}{a}\right)$

Find Minimum from the center of the diffraction pattern 

We can use this equation to find the value of the term$\left(\frac{\lambda L}{a}\right)$; note that we don't need to know the values of$a$, $\lambda$or$L$individually; instead, we can use this equation,

You only need to understand $\left(\frac{\lambda L}{a}\right)$, and you'll see why in a minute.

$$\begin{gathered} \frac{\lambda L}{a}=\frac{y_{\max ,1}}{1.43} \\ =\frac{26\mathrm{mm}}{1.43} \\ =18.2\mathrm{mm} \end{gathered}$$

The following reaction now describes the position of the initial minimum.

$y_p=\frac{p\lambda L}{a}p=1,2,3,\dots$

We have the value of $\frac{\lambda L}{a}$, and$p=1$. Hence

$$\begin{gathered} y_1=\frac{\lambda L}{a} \\ =18.2\mathrm{mm} \end{gathered}$$