Overview
This lecture covers how to calculate the wavelength and frequency of photons using mathematical formulas, with unit conversions and example problems.
Calculating Wavelength from Frequency
- Use the formula ( c = \lambda \nu ), where ( c ) is speed of light, ( \lambda ) is wavelength, ( \nu ) is frequency.
- To find wavelength: ( \lambda = \frac{c}{\nu} ).
- The speed of light (( c )) = ( 3 \times 10^8 ) meters/second.
- Example: For a frequency of ( 2.5 \times 10^{12} ) Hz, wavelength = ( 1.2 \times 10^{-4} ) meters.
- Convert meters to micrometers: ( 1.2 \times 10^{-4} ) m = 120 micrometers.
Calculating Frequency from Wavelength
- Use ( \nu = \frac{c}{\lambda} ).
- Example: For ( \lambda = 1.5 \times 10^{-8} ) m, frequency = ( 2 \times 10^{16} ) Hz.
- Always convert wavelength to meters before using the formula.
Unit Conversions for Wavelength
- Nanometers to meters: 1 nm = ( 10^{-9} ) meters.
- Example: ( 350 ) nm = ( 350 \times 10^{-9} ) meters.
- Frequency for ( 350 ) nm photon = ( 8.57 \times 10^{14} ) Hz.
Frequency and Wavelength Relationships
- As frequency increases, wavelength decreases (inverse relationship).
- As wavelength increases, frequency decreases.
Calculating with Different Units
- Megahertz to hertz: 1 MHz = ( 10^6 ) Hz.
- Example: 95 MHz = ( 95 \times 10^6 ) Hz; wavelength = ( 3.16 ) meters.
Calculating Frequency or Wavelength from Energy
- Energy-frequency relation: ( E = h\nu ), where ( h ) is Planck’s constant (( 6.626 \times 10^{-34} ) J·s).
- Frequency: ( \nu = \frac{E}{h} ).
- Example: For ( 3.5 \times 10^{-18} ) J, frequency = ( 5.28 \times 10^{15} ) Hz.
- To find wavelength from energy: 1) calculate frequency, 2) use ( \lambda = \frac{c}{\nu} ).
- Example: ( 4.3 \times 10^{-19} ) J → frequency ( 6.49 \times 10^{14} ) Hz → wavelength ( 4.62 \times 10^{-7} ) m or 462 nm.
Key Terms & Definitions
- Wavelength (( \lambda )) — The distance between two consecutive peaks of a wave, in meters.
- Frequency (( \nu )) — Number of wave cycles per second, measured in hertz (Hz).
- Speed of Light (( c )) — ( 3 \times 10^8 ) meters per second in a vacuum.
- Planck’s Constant (( h )) — ( 6.626 \times 10^{-34} ) joule·seconds.
- Photon — A particle of light with energy ( E = h\nu ).
Action Items / Next Steps
- Practice converting units between meters, nanometers, micrometers, and hertz.
- Solve additional problems using ( c = \lambda \nu ) and ( E = h\nu ).
- Remember to always use SI units in formulas unless specified.