1. Laser beam with wavelength 632.8 nm is aimed perpendicularly at opaque screen with two identical slits on it, positioned horizontally, and close enough so that both of them fall in the beam cross section. By shifting vertically the screen, each slits can be illuminated independently, allowing the other to be disregarded. On an observation screen positioned 1m further from the opaque screen, the diffraction patterns from the independent slit illumination were found identical, with minima 6.5 mm apart. The interference pattern maxima from simultaneous illumination of both slits were 0.53 mm apart. What are the double slit characteristics (slit width, slit separation)

Let's determine the slit characteristics in two steps:

First, we will determine the slit width

(b) using the single-slit diffraction pattern data, and then, Second, we will determine the slit separation

(d) using the double-slit interference pattern data.

1. To calculate the width of the slits (b), we need to use the formula for the minima of a single-slit diffraction pattern, which is given by:

\[y_m = \frac{(m + 0.5) \cdot \lambda \cdot L}{b}\]

Where: \(y_m\) = Position of the m-th minimum on the screen \(m\) = Order of the minimum (1st minimum, 2nd minimum, etc.) \(\lambda\) = Wavelength of the laser light \(L\) = Distance between the slit and the observation screen \(b\) = Slit width

For the first-order minimum (m = 1), and since the given distance between adjacent minima is 6.5 mm, we can imply \(2 \cdot y_m = 6.5 \, \text{mm}\) (because \(y_m\) is measured from the center of the pattern to the m-th minimum and the distance between consecutive minima would comprise two m values).

Let's solve for \(b\): \[2 \cdot y_m = \frac{(2m + 1) \cdot \lambda \cdot L}{b}\]

Given that \(\lambda = 632.8 \, \text{nm} = 632.8 \times 10^{-9} \, \text{m}\), and \(L = 1 \, \text{m}\), we get: \[6.5 \times 10^{-3} \, \text{m} = \frac{(2 \cdot 1 + 1) \cdot 632.8 \times 10^{-9} \, \text{m} \cdot 1 \, \text{m}}{b}\]

Solving for \(b\), we find that: \[b = \frac{3 \cdot 632.8 \times 10^{-9} \, \text{m}}{6.5 \times 10^{-3} \, \text{m}} \approx 0.292 \times 10^{-3} \, \text{m} = 0.292 \, \text{mm}\]

Now that we have the slit width, we can determine the slit separation (d) using the interference pattern results.

2. The maxima in a double-slit interference pattern can be found using the equation:

\[y_n = \frac{n \cdot \lambda \cdot L}{d}\]

Where \(y_n\) is the position of the nth maximum on the observation screen and \(d\) is the center-to-center separation distance between the slits.

Since the distance between adjacent maxima is given to be 0.53 mm: \[y_{n+1} - y_n = \frac{\lambda \cdot L}{d}\]

Given \(y_{n+1} - y_n = 0.53 \times 10^{-3} \, \text{m}\), solving for \(d\): \[d = \frac{\lambda \cdot L}{y_{n+1} - y_n} = \frac{632.8 \times 10^{-9} \, \text{m} \cdot 1 \, \text{m}}{0.53 \times 10^{-3} \, \text{m}} \approx 1.194 \times 10^{-3} \, \text{m} = 1.194 \, \text{mm}\]

So, the slit width is approximately 0.292 mm and the slit separation is approximately 1.194 mm.

Light is a form of electromagnetic radiation and behaves both as a wave and as a particle, which is known as wave-particle duality. - A key feature of wave behavior is interference, which involves the addition of wave amplitudes at a point in space. With coherent light (such as that from a laser), interference patterns form when light passes through two slits and overlaps on a screen. - In a single-slit diffraction pattern, the wave nature of light causes it to bend around the edges of the slit and interfere with itself, creating a pattern of bright and dark regions on a screen. - The central bright fringe in a single-slit diffraction pattern is flanked by dark regions called minima where destructive interference occurs. The angles (or positions) of these minima depend on the wavelength of light and the width of the slit. - A double-slit experiment is similar to the single-slit experiment, but with two slits, the overlapping light waves that emerge from the slits interfere with each other. Constructive interference leads to the bright fringes (maxima), while destructive interference causes the dark fringes. - By examining the interference and diffraction patterns generated from such experiments, one can determine the various characteristics of the slits used, such as their width and separation distance.

These experiments provided crucial evidence for the wave-like behavior of light and have important applications in fields like optical engineering and quantum physics.