The Photoelectric Effect Revision Notes

In these revision notes for The Photoelectric Effect, we cover the following key points:

• Why it is not always the same thing doing things in pieces and all at once?
• What is the Photoelectric Effect?
• What are the Laws of Photoelectric Effect? How were they discovered first?
• What is the stopping voltage? Why is it applied?
• Why the theoretical explanation of Photoelectric Effect was impossible within the framework of classical physics? How did Einstein overcome this issue?
• What is the work function? Where does it depend on?
• What is the Einstein's Equation of Photoelectric Effect?
• What is the theoretical explanation of the Laws of Photoelectric Effect?
• What is the quantic detachment caused by photoelectric effect?
• What is the intensity of this saturation current?

The Photoelectric Effect Revision Notes

The photoelectric effect is the phenomenon of electrons detachment from the surface of a metal when light falls on this metal surface. This phenomenon was first discovered by Hertz, in 1887. It occurs when light rays (photons) strike the surface of a metal. Thus, when frequency of photon is high enough to overcome the opposition made by metal atoms, an electron is detached from metal.

The increase of voltage in the circuit in which the metal cathode (and anode) are connected (starting from zero), is accompanied with the increase of current up to a maximum value I_sat, at which the saturation regime is set. This saturation current does not increase anymore with the increase in voltage as the number of electrons detached from cathode reach immediately the anode for higher voltages applied. The value of saturation current depends on the light flux and it is higher from greater fluxes.

There is some current flowing through the circuit even when the voltage in the circuit is zero, albeit very small. This phenomenon occurs because some electrons detached from cathode move directly towards anode, closing thus the circuit. To prevent this undesired current, a (negative) stopping voltage ΔV₀ is applied in the circuit by changing the polarities of electrodes. For higher negative values of stopping voltages, the current in the circuit remains zero.

The four laws of photoelectric effect are:

1. Photoelectric effect occurs only when the frequency f of the incident light is greater or at least equal to a frequency f₀, which is a characteristic of the metal used. This characteristic frequency is known as the threshold frequency of photoelectric effect. Mathematically, we write
   f ≥ f₀ ⟹ photoelectric effect does occur
2. The stopping voltage ΔV₀ depends only by the frequency f of the incident light in a proportional fashion. Mathematically, we write:
   ∆V₀ ∝ f
3. The number of photoelectrons detached from the metal plate in every second is proportional to the light flux falling on the cathode surface. Mathematically, we have:
   N(e)/t ∝ Φ/A
4. The photoelectric effect is a phenomenon that practically does not have any inertia. It occurs simultaneously with the light incidence on the cathode surface.

The phenomenon of photoelectric effect was explained by Einstein in 1905, who accepted as true the Planck's hypothesis on the discrete nature of light. Einstein elaborated further this idea by asserting that light is emitted, propagated and absorbed in specific portions (quanta) of energy. He supported the idea of the discrete nature of light and that light is composed by microscopic particles (photons) having the energy E_photon = h · f and impulse p = E / c. The Einstein's Equation of Photoelectric Effect derived from the law of conservation of energy, is

or

where Φ is known as the work function and it represents the minimum work necessary to detach the electron from metal and KE is the kinetic energy of electron after leaving the metal. The kinetic energy of electron in Einstein's Equation of Photoelectric Effect represents the maximum kinetic energy of electron, i.e. when applying this equation we assume the electron is on the surface of metal and not inside it; otherwise the kinetic energy of electron is smaller.

The condition for the photoelectric effect to occur is that

or

The quantity Φ/f depends on the type of metal and it represents the threshold frequency for the photoelectric effect to occur. We denote this threshold frequency by f₀, so we have:

During the stoppage of photoelectrons by the electric field of stopping voltage, the work done by the electric forces is equal to the kinetic energy of the fastest photoelectrons, i.e.

where e is the elementary charge or the charge of electron (e = -1.6 × 10^-19C) and ΔV₀ is the stopping voltage. Thus, we have

Therefore, we obtain for the stopping voltage:

From the above formula, we see that there is a linear relationship between the stopping voltage and light frequency, i.e. the stopping voltage increases with the increase of light frequency (2nd Law of Photoelectric Effect).

Photoelectric effect is a quantic phenomenon and as all the other quantic phenomena it is characterized by the probability of events occurrence. If we denote by N_ph the number of incident photons on the metal surface in one second and by Ne the number of photoelectrons produced in the same time, we have

This ratio (known as "quantic detachment caused by photoelectric effect") shows the probability for the photoelectric effect to occur. It is denoted by α and is a dimensionless quantity like all types of probability. Thus, we have

If α = 0, this means no electrons are detached from the surface of metal as the energy of all incident photons has been smaller than the work function.

If α = 1, this means each of the photons has detached one electron from metal. Theoretically, this occurs when all photons have a higher energy than work function. However, practically, this is impossible.

Photoelectrons that detach from cathode and reach the anode produce the photocurrent, which reaches the saturation value Is for specific values of accelerating voltage. The intensity of this saturation current is

The number of photons incident on the cathode in every second is determined through the light flux Φ, and is given by

Hence, the saturation current is proportional to the light flux, as

This outcome represents the third law of photoelectric effect.

The fourth law states that photoelectric effect is a phenomenon without inertia. This is evident, based on the fact that the photon-electron interaction practically occurs at instant.