Photon Wavelength and Frequency Calculations

Oct 10, 2024

Lecture Notes on Photon Wavelength and Frequency

Key Concepts

  • Speed of Light (c):
    • Constant value: (3 \times 10^8) meters/second.
  • Wavelength (λ):
    • Measured in meters.
  • Frequency (ν):
    • Measured in hertz (Hz).

Important Equations

  1. Wavelength and Frequency Relation:
    • (c = \lambda \times \nu)
    • Rearranged to find wavelength: (\lambda = \frac{c}{\nu})
    • Rearranged to find frequency: (\nu = \frac{c}{\lambda})
  2. Energy and Frequency Relation (Planck's Equation):
    • (E = h \times \nu)
    • (h = 6.626 \times 10^{-34}) joules (\cdot) seconds.

Problem Solving

Problem 1: Calculate Wavelength Given Frequency

  • Given: Frequency = (2.5 \times 10^{12}) Hz.
  • Solution:
    1. Use equation: (\lambda = \frac{c}{\nu}).
    2. Substitute values: (\lambda = \frac{3 \times 10^8}{2.5 \times 10^{12}}).
    3. Result: (\lambda = 1.2 \times 10^{-4}) meters.
    4. Convert to micrometers: (1.2 \times 10^{2}) micrometers (120 micrometers).

Problem 2: Calculate Frequency Given Wavelength

  • Given: Wavelength = (1.5 \times 10^{-8}) meters.
  • Solution:
    1. Use equation: (\nu = \frac{c}{\lambda}).
    2. Substitute values: (\nu = \frac{3 \times 10^8}{1.5 \times 10^{-8}}).
    3. Result: (\nu = 2 \times 10^{16}) Hz.

Problem 3: Frequency with Wavelength in Nanometers

  • Given: Wavelength = 350 nm.
  • Solution:
    1. Convert nm to meters: (350 \times 10^{-9}) meters.
    2. Use equation: (\nu = \frac{c}{\lambda}).
    3. Result: (\nu = 8.57 \times 10^{14}) Hz.

Problem 4: Wavelength Given Frequency in Megahertz

  • Given: Frequency = 95 MHz.
  • Solution:
    1. Convert MHz to Hz: (95 \times 10^{6}) Hz.
    2. Use equation: (\lambda = \frac{c}{\nu}).
    3. Result: (\lambda = 3.16) meters.

Conceptual Understanding

  • Inverse Relationship:
    • Wavelength and frequency are inversely related:
      • As frequency increases, wavelength decreases.
      • As wavelength increases, frequency decreases.

Additional Problems

Energy to Frequency and Wavelength

  1. Problem: Determine frequency of a photon with energy (E = 3.5 \times 10^{-18}) joules.

    • Solution:
      • Use equation: (\nu = \frac{E}{h}).
      • Result: (\nu = 5.28 \times 10^{15}) Hz.

  2. Problem: Determine wavelength from energy (E = 4.3 \times 10^{-19}) joules.

    • Solution:
      1. Calculate frequency: (\nu = \frac{E}{h}).
      2. Calculate wavelength: (\lambda = \frac{c}{\nu}).
      3. Convert result to nanometers: 462 nm.

These notes cover calculations involving photon wavelength and frequency, including conversions between units and understanding the inverse relationship between frequency and wavelength.