← Quantization of Light and Energy

# The Photoelectric Effect

Saturday, January 22, 2022

## Experimental Setup

It was observed that when light shines on a metal surface, electrons (called photoelectrons) can be emitted. This is called the photoelectric effect.

The photoelectrons can be collected, producing a current, $i$, in a circuit. Applying a potential difference, $\Delta V$, across the gap can stop the electrons from having enough energy to cross the gap.

The stopping potential, $\Delta V_S$, is the smallest potential where photoelectrons cannot clear the gap. The energy involved is described as:

$K_{max}=eV_S$

The belief, in classical wave theory, was that the energy from the electromagnetic wave was absorbed by the surface until enough energy was collected to exceed the binding energy of the electron.

### Work function, $\phi$

The work function, denoted $\phi$, of a material is the minimum amount of energy required to remove an electron

### Problems with classical wave theory

Classical wave theory predicts the following effects should be observed:

1. The maximum kinetic energy of the photoelectrons should be proportional to the intensity of the radiation
2. The photoelectric effect should occur for any wavelength or frequency
3. There should be a brief time delay in when the radiation begins and when the first electron is emitted from the surface

However, experimental observations produced the following results:

1. For set wavelengths and frequencies of light, the maximum kinetic energy of electrons is independent of the intensity of the light. Instead, the current is proportional to the intensity but not the stopping potential
2. The photoelectric effect does not occur if the frequency of light is below a certain threshold (called the cutoff frequency)
3. The first photoelectrons were emitted nearly instantly after the radiation began

## Quantum Theory of the Photoelectron Effect

The quantum theory of light proposed that light is delivered in discrete bundles, called quanta or photons.

### Photon energy and momentum

A photon's energy is determined by its wavelength or frequency:

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

Where $h$ is Planck's constant, approximately $6.626*10^{-34}m^2*kg/s$.

Photons carry linear momentum and energy but have no mass (or rest energy). And since photons travel at the speed of light, $E=pc$,

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

### Photons in the photoelectric effect

The photon theory explains all of the inconsistencies of the experimental observations of the photoelectric effect.

The cutoff frequency, $f_c$, which is the longest wavelength for which an electron will still be emitted, can be calculated as such:

$f_c=\frac{\phi}{h} \newline \lambda_c=\frac{hc}{\phi}$