# The Photoelectric Effect

The photoelectric effect was first observed by Heinrich Hertz in the 1880’s – yeah, the guy named after the rental car company (or was it the other way around?). Essentially, according to Wikipedia, he found that electrodes illuminated with ultra-violet light let off sparks more easily (where’d he get the UV?). This discovery led him to find that the light incident on the electrodes were “dislodging” the electrons in the metal, hence the EZ-Sparks. He also discovered that this was cool.

This is the experimental apparatus usually shown to beginners in quantum mechanics. The metal cathode is placed under an UV spotlight, and when you fiddle around with the intensity of the light, you find that the photo-current (as they call it) increases proportionally.

For ages we’ve known about something called the Work-Energy Theorem – which basically relates the work done to an object to its change in kinetic energy – and we’ve also known a little about how charges move in an Electric Field. If the electric potential difference from the cathode to the anode in the figure above is $V_{0}$, then the work done on an electron of charge e getting from the cathode to the anode is given by the product $eV_{0}$. By the Work-Energy Theorem, we can then derive an expression for the maximum kinetic energy of an electron getting that energy.

Now let’s think of a way to find out how to derive an expression for the energy of the light. The kinetic energy of the electron is being got from the energy of the light that “dislodges” it from the cathode. There’s some amount of energy required to do that though, which we might not know about, so for now we’ll call it the Work Function, $\phi$, representing the minimum amount of work (energy) needed to dislodge an electron. So whatever kinetic energy an electron in the above apparatus has, it will be whatever energy of light that hit it, minus the Work Function.

So, now… Why don’t we use what’s in our control to prevent the effect from happening? Let’s fiddle around and see what happens, like for instance, if there were a reverse potential difference, then it’s like rolling a ball up a hill. Eventually, make that hill tall enough, with the ball being thrown at the same speed as always, there’s no way in hell it’s getting to the top there. That precise reverse potential difference is called the stopping voltage, and it gives us insight.

Because it says that the maximum kinetic energy achievable at a certain light energy is equal to $eV_{stopping}$, hence $eV_{stopping}$ is equal in magnitude to the difference between the energy of the light and the work function.

${KE}_{max}=eV_{stopping}-\phi$

Do even more fiddling and you’ll find that this whole phenomena is dependent on the frequency of the light rays, not on the intensity, and that there exists a threshold frequency below which there will be no electrons being “dislodged”. This is kinda weird, if you thought light was made of waves instead of Photons, because even any wave of any frequency, no matter how dim, should be able to provide enough energy to an electron given enough time to do it – but this ain’t what we seeing.

Einstein did a little bit of mathematical fiddling, and derive the following expression for the energy of light: $E=hf$