Lecture Notes: Calculating Wavelength and Frequency of Photons

Key Concepts

• Photon's Wavelength and Frequency:
  • Wavelength (λ) and frequency (ν) are inversely related.
  • Speed of light (c) is a constant: (3 \times 10^8 \text{ m/s}).

Equations Used

• Basic Relation: ( c = \lambda \times \nu )
  • Rearranged for Wavelength: ( \lambda = \frac{c}{\nu} )
  • Rearranged for Frequency: ( \nu = \frac{c}{\lambda} )
• Energy Relation: Energy (E) and frequency relation: ( E = h \times \nu )
  • ( h ) is Planck's constant: (6.626 \times 10^{-34} \text{ J·s})

Problem Solutions

Problem 1: Calculate Wavelength

• Given: Frequency ( \nu = 2.5 \times 10^{12} \text{ Hz})
• Solution:
  • ( \lambda = \frac{3 \times 10^8}{2.5 \times 10^{12}})
  • Result: (1.2 \times 10^{-4} \text{ m})
  • Conversion to micrometers: (1.2 \times 10^2 = 120 \text{ micrometers})

Problem 2: Calculate Frequency

• Given: Wavelength (\lambda = 1.5 \times 10^{-8} \text{ m})
• Solution:
  • (\nu = \frac{3 \times 10^{8}}{1.5 \times 10^{-8}})
  • Result: (2 \times 10^{16} \text{ Hz})

Problem 3: Frequency with Nanometer Wavelength

• Given: Wavelength (\lambda = 350 \text{ nm})
• Conversion: (1 \text{ nm} = 10^{-9} \text{ m})
• Solution:
  • (\nu = \frac{3 \times 10^8}{350 \times 10^{-9}})
  • Result: (8.57 \times 10^{14} \text{ Hz})

Problem 4: Wavelength with Megahertz Frequency

• Given: Frequency (\nu = 95 \text{ MHz})
• Conversion: (95 \text{ MHz} = 95 \times 10^{6} \text{ Hz})
• Solution:
  • (\lambda = \frac{3 \times 10^8}{95 \times 10^6})
  • Result: (3.16 \text{ m})

Additional Insights

• Inverse Relationship:
  • As frequency (\nu) increases, wavelength (\lambda) decreases.
  • As wavelength (\lambda) increases, frequency (\nu) decreases.

Energy and Frequency

Problem: Calculate Frequency from Energy

• Given: Energy (E = 3.5 \times 10^{-18} \text{ J})
• Solution:
  • (\nu = \frac{3.5 \times 10^{-18}}{6.626 \times 10^{-34}})
  • Result: (5.28 \times 10^{15} \text{ Hz})

Problem: Wavelength from Energy

• Given: Energy (E = 4.3 \times 10^{-19} \text{ J})
• Steps:
  1. Calculate frequency: (\nu = \frac{4.3 \times 10^{-19}}{6.626 \times 10^{-34}})
     • Result: (6.49 \times 10^{14} \text{ Hz})
  2. Calculate wavelength: (\lambda = \frac{3 \times 10^8}{6.49 \times 10^{14}})
     • Result: (4.62 \times 10^{-7} \text{ m})
  3. Convert to nanometers: (4.62 \times 10^{-7} \text{ m} = 462 \text{ nm})

Conclusion

• Understanding the relationship between wavelength, frequency, and energy is crucial for solving physics problems related to photons.
• Proficiency in unit conversions is important for handling various problem contexts.