Energy of a Photon

• The energy of a photon is calculated using the Planck-Einstein relation, which links the particle-like energy to the wave-like frequency.

• Planck's Constant ($h$): A fundamental physical constant with a value of approximately $6.63 \times 10^{-34}$ J s.

• The relationship is expressed as:

$E = hf$

• In this formula, $E$ represents the energy in Joules (J) and $f$ represents the frequency of the radiation in Hertz (Hz).

• By incorporating the wave equation ($c = f\lambda$), photon energy can be expressed in terms of wavelength.

• Since frequency and wavelength are inversely proportional, the energy of a photon is also inversely proportional to its wavelength.

• The combined formula is:

$E = \frac{hc}{\lambda}$

• Here, $c$ is the speed of light ($3.00 \times 10^8$ m/s) and $\lambda$ is the wavelength in meters (m).

• This implies that shorter wavelengths (like UV or X-rays) carry significantly more energy per photon than longer wavelengths (like Infrared or Radio waves).

• Unit Conversion: Always ensure wavelength is converted from nanometers (nm) to meters (m) by multiplying by $10^{-9}$ before using the formula.

• Constants: Memorize the standard values for $h$ ($6.63 \times 10^{-34}$) and $c$ ($3.00 \times 10^8$), though they are usually provided on data sheets.

• Reasonableness Check: Photon energies for visible light are typically on the order of $10^{-19}$ Joules. If your answer is vastly different, check your powers of ten.

• Power vs. Energy: If a question asks for the number of photons emitted per second, divide the total power (Watts or J/s) by the energy of a single photon ($E$).

• Frequency vs. Wavelength: A common error is using the wavelength value in the $E = hf$ formula instead of the $E = hc/\lambda$ formula.

• Intensity Confusion: Students often think increasing the intensity (brightness) of light increases the energy of individual photons. It does not; it only increases the number of photons emitted per second.

• Planck's Constant Units: Ensure you are using Joules-seconds (J s) and not a different unit system that might lead to incorrect energy scales.