explain photoelectric emission in terms of photon energy and work function energy

Energy and Momentum of a Photon

What is a Photon? ⚡️

A photon is a tiny packet of light energy. Think of it like a single grain of sand on a beach – it carries a specific amount of energy and moves at the speed of light, \$c \approx 3.00 \times 10^8\,\text{m/s}\$.

Energy of a Photon

The energy of a photon depends on its frequency, \$\nu\$, or wavelength, \$\lambda\$:

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

where

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

Momentum of a Photon

Even though photons have no mass, they carry momentum:

\$p = \frac{E}{c} = \frac{h\nu}{c} = \frac{h}{\lambda}\$

This momentum is the reason light can push on mirrors or tiny particles (radiation pressure).

Photoelectric Effect – The Basics 📚

When light shines on a metal surface, electrons can be ejected. The key points are:

1. Each photon carries energy \$E_{\text{photon}} = h\nu\$.
2. Electrons in the metal need a minimum energy, called the work function, \$\phi\$, to escape.
3. If \$E_{\text{photon}} > \phi\$, the excess energy becomes the electron’s kinetic energy.
4. Electrons with \$E_{\text{photon}} \le \phi\$ are not emitted.

Energy Balance Equation

The relationship between photon energy, work function, and electron kinetic energy is:

\$E_{\text{photon}} = \phi + \frac{1}{2}mv^2\$

where \$m\$ is the electron mass and \$v\$ its speed after ejection.

Example Calculation 🚀

Suppose a metal has a work function \$\phi = 2.20\,\text{eV}\$ and is illuminated with light of wavelength \$\lambda = 400\,\text{nm}\$.

1. Convert wavelength to energy:

   \$E_{\text{photon}} = \frac{hc}{\lambda} = \frac{(4.1357 \times 10^{-15}\,\text{eV·s})(3.00 \times 10^8\,\text{m/s})}{400 \times 10^{-9}\,\text{m}} \approx 3.10\,\text{eV}\$

2. Find maximum kinetic energy:

   \$K{\max} = E{\text{photon}} - \phi = 3.10\,\text{eV} - 2.20\,\text{eV} = 0.90\,\text{eV}\$

Key Takeaways

• Photon energy increases with frequency and decreases with wavelength.
• Momentum of a photon is \$p = h/\lambda\$; it can transfer momentum to matter.
• In the photoelectric effect, only photons with enough energy to overcome the work function can eject electrons.
• Any excess photon energy becomes the kinetic energy of the emitted electron.