Photoelectric Effect

We have seen how the energy in incoming photons can be absorbed by electrons. Turns out if we hit an electron hard enough we can knock it out of orbit. When this occurs the electron gets a special name a photoelectron because it is an electron that was knocked out of orbit by a photon.

Energy Tax

In order for an electron to escape its orbit, the incoming photon has to pay an “energy tax”. This tax is called the work function (wf) and represents the lowest amount of energy required to eject an electron. Any energy beyond the work function goes to the electron causing it to speed away from the material surface.

Here we see conservation of energy at work. The energy in the photon is converted into the kinetic energy in the electron and into overcoming the attractive energy of the nucleus on the electron in question.

[latexpage]
\[KE_{photoelectron} = E_{photon} – wf
\]

Packing a Punch

Interestingly the energy of the individual photon matters far more than the brightness of the light. This seems counterintuitive, doesn’t turning up the light increase the amount of energy a piece of metal receives? Of course, it does, but to an electron only the energy of individual photons matter.

So more electrons will be ejected when we turn the light up. However, all of those electrons will end up with the same amount of energy as before. If we want to generate electrons with more energy and thus fasters speeds we need to increase the energy of the photon that knocked it out of place.

[latexpage]
Specifically
\[
E=hf
\]
or \[
E=\frac{hc}{\lambda}
\]

This means that as our frequency increases or as our wavelength decreases the energy of a photon will go up. So blue light with a frequency of 6.6×10¹⁴ Hz will generate much higher energy electrons than red light which has a frequency around 4.29×10¹⁴ Hz. However, red light with an intensity of 10 lux will eject far more photons than a blue light with an intensity of 3 lux.

If we want to quantify this more exactly we can utilize our above equation by replacing E of the photon with hf in order to calculate the overall energy a photoelectron has.

[latexpage]
\[KE_{photoelectron}=hf-wf
\]
or if we are only given the wavelength
\[KE_{photoelectron}=\frac{hc}{\lambda}- wf
\]

Furthermore, we can calculate the speed of an electron so long as we know how much kinetic energy is received by using the kinetic energy equation.

[latexpage]
\[
KE_{photoelectron}= \frac{1}{2}mv^2
\]

Who Cares?

When enough electrons leave the metal’s surface they generate a current. Sensors such as those found in smoke detectors harness the photoelectric effect to determine the presence or absence of smoke. When no smoke is present in the air a smoke detector bombards a metal plate with UV light causing photoelectrons to generate a current.

If smoke or other particles get in the way of the light beam and the metal plate then the current will stop. In response, the smoke detector starts wailing to alert us that there might be a fire or that your dinner is a bit overdone.