Energy of a Photon

• Planck's Relation: The energy of a photon is governed by the equation $E = hf$. This establishes that energy ($E$) is directly proportional to the frequency ($f$) of the radiation.

• Planck's Constant ($h$): This fundamental constant, approximately $6.63 \times 10^{-34} \text{ J s}$, acts as the proportionality factor that scales the frequency of an electromagnetic wave to its energy equivalent.

• Wave-Particle Duality: While light exhibits wave-like properties (interference and diffraction) during propagation, it exhibits particle-like properties (discrete energy transfer) when interacting with matter. The photon energy formula bridges these two descriptions.

Calculating Energy from Frequency

• Use the primary formula: $E = hf$
• Ensure frequency is in Hertz (Hz or $s^{-1}$) and energy will be calculated in Joules (J).

Calculating Energy from Wavelength

• Since wave speed $c = f\lambda$, we can substitute $f = \frac{c}{\lambda}$ into the energy equation to get: $E = \frac{hc}{\lambda}$
• This method is essential when the wavelength is known instead of the frequency. Note that energy is inversely proportional to wavelength.

Unit Conversion (J to eV)

• In atomic physics, Joules are often too large. The electron-volt (eV) is used instead, where $1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}$.
• To convert Joules to eV, divide the energy by $1.60 \times 10^{-19}$.

• Check Wavelength Units: Wavelengths are frequently given in nanometers (nm). Always convert to meters ($1 \text{ nm} = 10^{-9} \text{ m}$) before plugging into $E = \frac{hc}{\lambda}$ to avoid massive calculation errors.

• Proportionality Logic: If a question asks how energy changes when frequency doubles, remember $E \propto f$, so energy also doubles. If wavelength doubles, energy is halved ($E \propto \frac{1}{\lambda}$).

• Sanity Check: Visible light photons typically have energies in the range of $10^{-19} \text{ J}$ or $1.5$ to $3.5 \text{ eV}$. If your calculated value is significantly outside this range for visible light, re-check your powers of ten.

• The 'Intensity' Trap: Many students incorrectly assume that increasing the brightness (intensity) of light increases the energy of the photons. Intensity only increases the count of photons; only changing the color (frequency) changes the energy per photon.

• Inverse Relationship Errors: Students often forget that a larger wavelength means a smaller energy. Blue light (shorter $\lambda$) is more energetic than red light (longer $\lambda$).

• Constant Confusion: Ensure you do not confuse Planck's constant ($h$) with the speed of light ($c$) or the charge of an electron ($e$) when performing multi-step calculations.