Slits and Gratings

🔖 Topics

• Single slit apparatus
• Double slit apparatus
• Diffraction gratings
• Small angle approximation

🎯 Objectives

• Describe the origin of interference effects in slits and gratings
• Calculate the positions of maxima and minima in the single slit apparatus
• Calculate the positions of maxima and minima in the double slit apparatus
• Calculate the positions of maxima and minima in diffraction gratings
• Describe the cumulative effects of a double slit apparatus where the slits are given non-zero width.
• Use the small angle approximation to calculate the spacing between maxima or minima, wherever appropriate

📋 Sequence

• Constructive versus destructive interference
• Wave interference
• Double slit apparatus
  • Maxima: \(d sin\theta = m\lambda\) where \(m = 0, \pm 1, \pm 2, ...\)
  • Minima: \(d sin\theta = (m - \frac{1}{2})\lambda\) where \(m = \pm 1, \pm 2, ...\)
• Diffraction grating
  • Maxima: \(d sin\theta = m\lambda\) where \(m = 0, \pm 1, \pm 2, ...\)
  • Minima: \(d sin\theta = (m - \frac{1}{2})\lambda\) where \(m = \pm 1, \pm 2, ...\)
• Single slit apparatus
  • Minima: \(a sin\theta = m \lambda\) where \(m = \pm 1, \pm 2, \pm 3, ...\)
• Small angle approximation: \(sin\theta \approx tan\theta \approx \frac{y}{L}\)
• Example problems for double slits, single slits, and diffraction gratings

🖥️ Animations, Simulations, Activities

• Wave Interference PhET Simulation: https://phet.colorado.edu/sims/html/wave-interference/latest/wave-interference_all.html
• Double Slit Interference GeoGebra Simulation: https://www.geogebra.org/m/ynwgtfk8

📝 Practice Problems

1. A double slit apparatus has a slit separation of 0.20 mm and is 1.5 meters away from a screen. If I shine coherent light with a wavelength of 700 nm onto the screen, what are the positions of the central, first, second, and third maxima with respect to the center?
2. A diffraction grating has a slit separation of 0.10 mm. If I shine coherent light with a wavelength of 650 nm onto the apparatus and view the pattern on a screen, what is the maximum, theoretical number of maxima that I would see?
3. You set up a single slit apparatus and measure the third order minimum at an angle of \(4^\circ\). If the frequency of the light on the apparatus is \(6 \times 10^{14}\) Hz, what is the width of the slit?

✅ Partial Solutions

1. 0.0 cm, 0.5 cm, 1.0 cm, 1.6 cm
2. 307
3. 21.5 \(\mu m\)

📘 Connected Resources

• Giambattista, Alan, et al. College Physics With an Integrated Approach to Forces and Kinematics. 5th ed., McGraw-Hill Education, 2020.
• Tipler, Paul A., and Gene Mosca. Physics for Scientists and Engineers: With Modern Physics. 6th ed., W. H. Freeman, 2008.