# Why does the photoelectric effect depend on the frequency of the light?

In experiments to detect the photoelectric effect, a clean metal was irradiated by monochromatic light and electrons were emitted.

Why was monochromatic light used in the experiment, and why does the frequency of the light have to be above a threshold frequency?

## 3 Answers

The photoelectric effect is described by the following equation

$$E_\mathrm{max} = h\nu - \mathrm{WF_M}$$

where $E_\mathrm{max}$ is the maximum kinetic energy of the electron escaping from the metal surface, $\nu$ is the frequency of the incoming photon and $\mathrm{WF_M}$ is the workfunction for the particular metal. The kinetic energies of all electrons emitted are distributed from $0$ to $E_\mathrm{max}$.

Why was monochromatic light used in the experiment? Why is a proper frequency used and not any other frequency?

The experiment is typically performed by scanning through a continuous range of monochromatic wavelengths from lower to higher energy. At some specific wavelength, the observer will notice that electrons start to be emitted (the threshold). As the scan continues to wavelengths with even more energy, the emitted electrons will increase in kinetic energy. From the threshold energy and the above equation, the experimenter can determine the workfunction of the metal. If the experiment had been run with light containing many different wavelengths (non-monochromatic light), electrons would still be ejected, but you wouldn't know what the threshold wavelength was and you wouldn't be able to determine the workfunction of the metal being studied.

If you'd like more information on the photoelectric effect, here's a good, concise reference.

The physical background of the experiment is that inside the metal the electrons can occupy states up till a point. These states are more or less continuous, however the electrons fill these states till an energy, called Fermi level. This energy is lower than the energy of electron in vacuum, so the electron needs an extra energy to jump. If the Fermi energy was not lower than vacuum, the electrons would just leave the metal, and fly away. When PE happens and a photon is absorbed, and it gives enough energy, and electron can jump out.

The important point here is that a single photon gives all the energy to a single electron. In a classical theory, light is just wave, it has no reason to excite a specific electron. Also, if it is continuous wave, electrons could collect enough energy just "waiting longer" if the frequency of light is lower. But PE works this way, because light is quantized: one photon is not the same as two photons with half energy. To jump, the electron needs enough energy from a single photon, because the chance that it is hit by two photons is practically zero, and there may not be empty states along the way and so cannot just add up energy piece by piece.

Monochromatic light

You don't need monochromatic light. If you use monochromatic light, than all the photons correspond to a given energy, therefore easier to interpret the data, but it is just a practical requirement. If you use normal light, all kind of photos will come, some bellow threshold, some above, so all you see is the mixture of zillion energies.

Threshold energy

As I said, this is the whole point. The electron cannot just absorb 3 and a half photons to get enough energy. If you have enough energy from a photon, you see PE, if not, there is no PE. The threshold energy which tells you if the energy is enough.

The wave theory fails to explain the observations

Now, one might try to explain this photoelectric effect with waves of light: light waves impinge on the source plate. They give energy to electrons on the plate. The electrons fly off the source plate to the receptor. The wave theory predicts several things:

The more intense the light, the more energy the electrons will have when the fly off the plate. If the light is very feeble, one may have to expose the source plate for several seconds or minutes until enough waves strike it to knock electrons loose. Waves of any frequency ought to knock electrons free. Careful measurements in the lab, however, showed that these predictions were wrong, wrong, wrong.

The energy of the electrons does NOT depend on the intensity of the light. The electrons always appear AS SOON AS the light reaches the plate (though a feeble light produces only a few). NO electrons are produced if the frequency of the light waves is below a critical value.

Albert Einstien while Explaining Photo electric Effect assumed the following things.

1. Light is made up of photons.

2. One photon can only knock out 1 electron

So As per Planks Equation E = h(nu) and $E= mv^2/2$

so a specific frequency till which electrons are not ejected is called thresh hold frequency. $$W=hf$$ where W is the work function, h is Planck's constant, and f is the threshold frequency.

After that, the energy supplied to the electrons is then converted to kinetic energy

Thus Total Energy $$E = W + KE$$ where KE is just the kinetic energy of electron ejected.

Which means "The higher The frequency More will be the speed of ejected electron"

PS: - Monochromatic light is used so that there is no intermixing of frequencies!

The term monochromatic comes from the Greek words mono, meaning single, and chroma, meaning colour. So monochromatic light literally means light of one colour. In scientific terms, it means light of a single wavelength or frequency. Light is a term for the visible and near visible portions of electromagnetic radiation

ALSO,

ALBERT EINSTIEN RECEIVED NOBEL PRIZE FOR THIS EXPLANATION NOT FOR RELATIVITY!