Physics Lesson 19.2.3 - Explanation of the Laws of Photoelectric Effect

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Welcome to our Physics lesson on Explanation of the Laws of Photoelectric Effect, this is the third lesson of our suite of physics lessons covering the topic of The Photoelectric Effect, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.

Explanation of the Laws of Photoelectric Effect

From Einstein's Equation of Photoelectric Effect, it is obvious that this effect takes place only when photon's energy is greater than or equal to the work function (1st Law of Photoelectric Effect). Otherwise, electron could not be able to detach from metal surface. Therefore, the condition for the photoelectric effect to occur is that

hf ≥ Φ

f ≥ Φ/h

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:

f₀ = Φ/h

We can calculate the value of threshold frequency for every metal shown in the table of work function provided in the previous paragraph. For example, the threshold frequency for iron (Φ = 4.36 e_V,/sub>), is

f₀ = Φ/h
= 4.36 ∙ 1.6 × 10^-19 J/6.626× 10^-34 J · s
= 1.05 × 10¹⁵ Hz

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.

KE = e ∙ ∆V₀

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

h ∙ f = Φ + e ∙ ∆V₀

Therefore, we obtain for the stopping voltage:

∆V₀ = h ∙ f/e - Φ/e

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). This relationship is shown in the graph below.

Physics Tutorials: This image provides visual information for the physics tutorial The Photoelectric Effect

Photoelectric effect is a quantic phenomenon and as all the other quantic phenomena it is characterized by the probability of events occurrence. This means not all incident photons can cause photoelectric effect on a metal by detaching an electron from its surface. Some incident photons are absorbed by atoms of metals without producing any photoelectric effect while some other photons are reflected back by the metal surface. Thus, a photon can produce one or no photoelectrons when it falls on a metal.

If we denote by Nph 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

N_e/N_ph ≤ 1

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

α = N_e/N_ph

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

I_s = e ∙ N_e
= e ∙ α ∙ N_ph

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

N_ph = Φ/h ∙ f

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

I_s = e ∙ α ∙ (Φ/h ∙ f)

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.

In this way, based on the quantum hypothesis on the particle nature of light, we managed to explain the laws of photoelectric effect found experimentally.

Example 2

What is the stopping voltage when a 500 nm photon falls on a sodium cathode (Φ = 2.26 eV)? Take the charge of electron e = 1.6 × 10^-19 C.

Solution 2

Clues:

λ = 500 nm = 5 × 10-7 m
Φ = 2.26 eV = 2.26 × 1.9 × 10^-19
J = 3.616 × 10^-19 J
e = 1.6 × 10^-19 C
(c = 3 × 10⁸ m/s)
(h = 6.626 × 10^-34 J · s)

First, we calculate the frequency of photon. Thus, giving that

c = λ ∙ f

we obtain for the frequency f of photon:

f = c/λ
= 3 × 10⁸ m/s/5 × 10^-7 m
= 0.6 × 10¹⁵ Hz
= 6 × 10¹⁴ Hz

Now, using the equation

∆V₀ = h ∙ f/e-Φ/e

∆V₀ = h ∙ f - Φ/e

we obtain for the stopping voltage on the sodium cathode for the given frequency:

∆V₀ = (6.626 × 10^-34 J ∙ s) ∙ (6 × 10¹⁴ Hz)-(3.616 × 10^-19 J)/1.6 × 10^-19 C
= 3.976 × 10^-19J - 3.616 × 10^-19 J/1.6 × 10^-19 C
= 0.225 V

You have reached the end of Physics lesson 19.2.3 Explanation of the Laws of Photoelectric Effect. There are 3 lessons in this physics tutorial covering The Photoelectric Effect, you can access all the lessons from this tutorial below.

More The Photoelectric Effect Lessons and Learning Resources

Modern Physics Learning Material
Tutorial ID	Physics Tutorial Title	Revision Notes	Revision Questions
19.2	The Photoelectric Effect
Lesson ID	Physics Lesson Title	Lesson	Video Lesson
19.2.1	Laws of the Photoelectric Effect
19.2.2	Einstein's Equation of Photoelectric Effect. The Particle Nature of Light
19.2.3	Explanation of the Laws of Photoelectric Effect

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