Photon Wavelength and Frequency Problems

Introduction

• Focus on calculating wavelength and frequency of photons.

Problem 1: Wavelength Calculation

• Given: Frequency = 2.5 × 10²² Hz
• Equation: c = λ × ν (where c = speed of light, λ = wavelength, ν = frequency)
• Rearranged: λ = c / ν
• Values:
  • c = 3 × 10⁸ m/s
  • ν = 2.5 × 10¹² Hz
• Calculation:
  • λ = (3 × 10⁸ m/s) / (2.5 × 10¹² Hz)
  • Result: λ = 1.2 × 10⁻⁴ m
• Conversion to Micrometers:
  • 1 micrometer = 10⁻⁶ m
  • λ = 1.2 × 10⁻⁴ m × 10⁶ = 1.2 × 10² micrometers
  • Result: λ = 120 micrometers

Problem 2: Frequency Calculation

• Given: Wavelength = 1.5 × 10⁻⁸ m
• Equation: ν = c / λ
• Calculation:
  • ν = (3 × 10⁸ m/s) / (1.5 × 10⁻⁸ m)
  • Result: ν = 2 × 10¹⁶ Hz

Problem 3: Frequency from Wavelength in Nanometers

• Given: Wavelength = 350 nm
• Conversion: 350 nm = 350 × 10⁻⁹ m
• Calculation:
  • ν = c / λ
  • ν = (3 × 10⁸ m/s) / (350 × 10⁻⁹ m)
  • Result: ν = 8.57 × 10¹⁴ Hz

Problem 4: Wavelength from Frequency (Megahertz)

• Given: Frequency = 95 MHz
• Conversion: 95 MHz = 95 × 10⁶ Hz
• Calculation:
  • λ = c / ν
  • λ = (3 × 10⁸ m/s) / (95 × 10⁶ Hz)
  • Result: λ = 3.16 m

Relationship between Wavelength and Frequency

• Wavelength and frequency are inversely related:
  • As frequency increases, wavelength decreases and vice versa.

Energy and Frequency Calculation

• Equation: E = h × ν (where E = energy, h = Planck's constant)
• Planck's Constant: h = 6.626 × 10⁻³⁴ J·s
• Given: E = 3.5 × 10⁻¹⁸ J
• Calculation for Frequency:
  • ν = E / h
  • ν = (3.5 × 10⁻¹⁸ J) / (6.626 × 10⁻³⁴ J·s)
  • Result: ν = 5.28 × 10¹⁵ Hz

Problem 5: Wavelength from Energy

• Given: E = 4.3 × 10⁻¹⁹ J
• Calculate Frequency:
  • ν = E / h
  • ν = (4.3 × 10⁻¹⁹ J) / (6.626 × 10⁻³⁴ J·s)
  • Result: ν = 6.49 × 10¹⁴ Hz
• Calculate Wavelength:
  • λ = c / ν
  • λ = (3 × 10⁸ m/s) / (6.49 × 10¹⁴ Hz)
  • Result: λ = 4.62 × 10⁻⁷ m
• Convert to Nanometers:
  • λ = 4.62 × 10⁻⁷ m × 10⁹ = 462 nm

Conclusion

• Understanding the relationships between wavelength, frequency, and energy is crucial in photon calculations.