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, $u$, or wavelength, $\lambda$: $$E = hu = \frac{hc}{\lambda}$$ 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: $$p = \frac{E}{c} = \frac{hu}{c} = \frac{h}{\lambda}$$ 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 $E_{\text{photon}} = hu$.
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.