# Photoelectric effect

The photoelectric effect was well known by the 1900s, but there wasn’t yet an explanation for it. Using a UV lamp, it was possible to neutralize the static charge on some negatively charged metal plates. To work, it required

• negatively charged plate;
• only certain metals;
• fresh surfaces (no oxidation);
• ultraviolet radiation, not visible light.

The plates lost their charge because the ultraviolet rays caused electrons to electrons to fly away from the plate. These electrons are called photoelectrons. An interesting experiment with the photoelectric effect was to use UV to propel electrons across a vacuum from one electrode to another, thus completing a circuit and causing photocurrent to flow:

The variables in this experiment were the type of metal (of the target electrode), the frequency of the UV source, and the intensity or brightness of the ultraviolet light. In the diagram above, their is a variable power supply. Even without applying any voltage, the UV rays cause photoelectrons to shoot off the target plate and bounce around in the tube. When a voltage is applied, the target becomes negatively charged and the collector becomes positively charged, and the flying electrons are drawn towards the collector by electric forces.

We can stop the photoelectrons by applying a voltage with opposite polarity (this voltage is called the retarding potential). The collector plate is now positive, and it repels the flying electrons. But by how much? As it turns out, the current decreases steadily as we apply more voltage—until we reach the cut-off potential, where current drops abruptly:

Once we reach $\displaystyle V_{0}$, even the fastest electrons are prevented from reaching the collector electrode. This was helpful for determining the maximum kinetic energy (and from that, speed) of the electrons.

Let’s return to those other variables. For each metal, there was a different threshold below which no current flowed at all, denoted by $\displaystyle f_{0}$. After the frequency passes $\displaystyle f_{0}$, higher frequencies cause the photoelectrons to move faster, which in turn increases value of $\displaystyle V_{0}$ because they are harder to stop. As the intensity of the ultraviolet rays increases, more photoelectrons get released—but they do not move any faster or slower. In all cases, the electrons are released instantaneously. Varying frequency and intensity changes the above graph like this:

Frequency controls $\displaystyle E_{\text{k}}$ and therefore speed and $\displaystyle V_{0}$; intensity controls current. This is the opposite of the wave model’s predictions, and no one could explain it—until Einstein did in 1905. He used Planck’s idea of photon energy, $\displaystyle E = h f$. The energy of a photon can only be absorbed by a single electron; there is no sharing. This electron may then have enough energy to break free from the metal. When $\displaystyle f{>} f_{0}$, there is enough energy to break free and some excess. This extra energy is converted to kinetic energy, shooting the photoelectron into the vacuum. As the intensity of the ultraviolet light increases, there are more photons hitting the target every second, therefore more electrons are flowing and the current increases.

Some electrons are right on the surface and move away from it; others are a bit farther in and have to give up some $\displaystyle E_{\text{k}}$ to fight their way through; others still head inward first, then rebound and go the other way, losing $\displaystyle E_{\text{k}}$ in the process. The $\displaystyle E_{\text{k}}$ of the photoelectron ranges from zero to some maximum value, $\displaystyle E_{\text{k,max}}$. The fastest electrons are the ones that require a voltage of $\displaystyle V_{0}$ to be fully repelled. With this information, Einstein developed the formula

$\displaystyle E_{\text{k,max}} = h f - W$,

where $\displaystyle W$ is the work function, representing the energy needed for an electron to escape the electrode. The work function depends on the metal in the target electrode, and it can be measured in joules (J) or in electron-volts (eV). One electron-volt is equal to 1.60 × 10−19 J.

Since $\displaystyle f_{0}$ is the frequency that allows an electron to break free but gives no extra energy to move, it will result in $\displaystyle E_{\text{k,max}} = 0$. This gives us another equation, which we can use to define $\displaystyle W$ as well as a way to find $\displaystyle f_{0}$:

$\displaystyle W = h f_{0}$.