Question

part a
the third - order bright fringe of 550 nm light is observed at an angle of 27° when the light falls on two narrow slits
how far apart are the slits?
express your answer to two significant figures and include the appropriate units
(d=) value units

Explanation:

Step1: Recall Double - Slit Formula

The formula for the position of bright fringes in a double - slit experiment is \(d\sin\theta = m\lambda\), where \(d\) is the slit separation, \(\theta\) is the angle of the fringe, \(m\) is the order of the fringe, and \(\lambda\) is the wavelength of the light. We need to solve for \(d\), so we can rearrange the formula to \(d=\frac{m\lambda}{\sin\theta}\).

Step2: Identify Given Values

We are given that \(m = 3\) (third - order bright fringe), \(\lambda=550\space nm=550\times 10^{-9}\space m\), and \(\theta = 27^{\circ}\).

Step3: Substitute Values into the Formula

First, calculate \(\sin(27^{\circ})\approx0.454\). Then substitute \(m = 3\), \(\lambda = 550\times 10^{-9}\space m\) and \(\sin\theta=0.454\) into the formula for \(d\):

\(d=\frac{3\times550\times 10^{-9}\space m}{0.454}\)

\(d=\frac{1650\times 10^{-9}\space m}{0.454}\approx3.634\times 10^{-6}\space m\)

Step4: Round to Two Significant Figures

Rounding \(3.634\times 10^{-6}\space m\) to two significant figures gives \(3.6\times 10^{-6}\space m\) or \(3.6\space\mu m\) (since \(1\space\mu m = 10^{-6}\space m\)).

Answer:

\(d = 3.6\space\mu m\) (or \(3.6\times 10^{-6}\space m\))