understand that photoelectrons may be emitted from a metal surface when it is illuminated by electromagnetic radiation

Energy and Momentum of a Photon

What is a Photon? ⚡️

A photon is a tiny, indivisible packet of light. Think of it as a “light‑ball” that carries energy and can also push on objects, just like a tiny bowling ball can knock over pins. Photons have no mass, but they do have momentum, which is why they can exert pressure on surfaces (the radiation pressure we see in solar sails).

Energy of a Photon 🌞

The energy of a photon depends on its frequency (or wavelength). The two most common ways to write it are:

  • \$E = h \nu\$  (energy = Planck constant × frequency)
  • \$E = \dfrac{hc}{\lambda}\$  (energy = Planck constant × speed of light ÷ wavelength)

Where:

  • \$h = 6.626 \times 10^{-34}\,\text{J·s}\$ (Planck constant)
  • \$c = 3.00 \times 10^8\,\text{m/s}\$ (speed of light)
  • \$\nu\$ = frequency (Hz)
  • \$\lambda\$ = wavelength (m)

Momentum of a Photon 🚀

Even though photons have no mass, they carry momentum given by:

\$p = \dfrac{h}{\lambda}\$

This tiny push is enough to move a solar sail across space or to create a measurable force on a delicate instrument in a laboratory.

The Photoelectric Effect 🔬

When light shines on a metal surface, it can knock electrons out of the metal. This happens only if the photon’s energy is large enough to overcome the metal’s work function (the energy needed to free an electron). The relationship is:

\$E{\text{photon}} = \phi + KE{\text{max}}\$

Where:

  • \$\phi\$ = work function of the metal (J)
  • \$KE_{\text{max}}\$ = maximum kinetic energy of the ejected electron (J)

If \$E_{\text{photon}} < \phi\$, no electrons are emitted, no matter how bright the light is. This explains why a very bright but low‑frequency (red) light can fail to produce photoelectrons, whereas a dim but high‑frequency (ultraviolet) light can.

Analogy: The “Light‑Ball” Toss

  • Imagine a ball (photon) thrown at a stack of coins (electrons in a metal). If the ball is heavy (high energy), it can knock a coin out of the stack. If the ball is light (low energy), it just bounces off.
  • The ball’s speed (frequency) and size (wavelength) determine how much energy it carries.
  • Even a light ball can push on a very light object (momentum of photon).

Key Equations Summary 📚

QuantityFormulaUnits
Energy of a photon\$E = h\nu = \dfrac{hc}{\lambda}\$Joules (J)
Momentum of a photon\$p = \dfrac{h}{\lambda}\$kg·m/s
Photoelectric condition\$E_{\text{photon}} \ge \phi\$J
Maximum kinetic energy of photoelectron\$KE{\text{max}} = E{\text{photon}} - \phi\$J

Quick Check Questions ??

  1. What happens to the photoelectric current if you increase the light intensity but keep the frequency below the threshold? 🤔
  2. How does the wavelength of a photon affect its momentum? 📏
  3. Why can a high‑frequency, low‑intensity light still produce photoelectrons, while a low‑frequency, high‑intensity light cannot? 🔍