Photoelectric Effect Slide

The Photoelectric Effect: Definition and Experimental Setup

Late in the nineteenth century, a series of experiments revealed that electrons are emitted from a metal surface when light of sufficiently high frequency falls upon it. This phenomena is known as the photoelectric effect.

Experimental Setup and Basic Function

An evacuated tube contains two electrodes connected to a source of variable voltage. The metal plate whose surface is irradiated acts as the anode. Some of the photoelectrons that emerge from this surface have enough energy to reach the cathode despite its negative polarity, and they constitute the measured current.

Image/Diagram: Experimental observation of the photoelectric effect. (The diagram shows an evacuated quartz tube with light striking the positive electrode (anode), causing electrons to travel to the negative electrode (cathode). The circuit includes a variable resistor, battery, voltmeter (V), and ammeter (A)).

The slower photoelectrons are repelled before they get to the cathode. When the voltage is increased to a certain value $V_{0}$ , of the order of several volts, no more photoelectrons arrive, as indicated by the current dropping to zero. This extinction voltage ( $V_{0}$ ) corresponds to the maximum photoelectron kinetic energy.

Conflict with Classical (EM) Theory

The existence of the photoelectric effect is not surprising on a classical basis, as light waves carry energy. It was thought that some of the energy absorbed by the metal might concentrate on individual electrons and reappear as their kinetic energy, similar to water waves dislodging pebbles from a beach. However, three experimental findings show that no such simple explanation is possible based on the classical electromagnetic (EM) theory of light.

1. Time Delay (Observation vs. Prediction)

Within the limits of experimental accuracy (about $1 0^{- 9}$ s), there is no time interval between the arrival of light at a metal surface and the emission of photoelectrons.

Classical Prediction: However, because the energy in an EM wave is supposed to be spread across the wavefronts, a period of time should elapse before an individual electron accumulates enough energy (several eV) to leave the metal. A detectable photoelectron current results when $1 0^{- 6}$ W/m $^{2}$ of EM energy is absorbed by a sodium surface. At this rate, considering a layer of sodium 1 atom thick and 1 m $^{2}$ in area containing about $1 0^{19}$ atoms, each atom receives energy at an average rate of $1 0^{- 25}$ W. At this slow rate, over a month would be needed for an atom to accumulate energy of the magnitude that photoelectrons from a sodium surface are observed to have.

2. Intensity and Energy

A bright light yields more photoelectrons than a dim one of the same frequency, but the electron energies remain the same.

Classical Prediction: The EM theory of light, on the contrary, predicts that the more intense the light, the greater the energies of the electrons.

Image/Graph: Shows Photoelectron current versus Retarding potential (V) where frequency ( $v$ ) is constant. The graph shows curves corresponding to intensity $I$ , $2 I$ , and $3 I$ all converging to the same stopping potential $V_{0}$ .

Conclusion from Graph: Photoelectron current is proportional to light intensity $I$ for all retarding voltages. The stopping potential $V_{0}$ , which corresponds to the maximum photoelectron energy, is the same for all intensities of light of the same frequency.

3. Frequency and Energy (The Critical Frequency)

The higher the frequency of the light, the more energy the photoelectrons have. Blue light results in faster electrons than red light.

Observation: At frequencies below a certain critical frequency $v_{0}$ , which is characteristic of each particular metal, no electrons are emitted. Above $v_{0}$ , the photoelectrons range in energy from 0 to a maximum value that increases linearly with increasing frequency. This observation, also, cannot be explained by the EM theory of light.

Image/Graph: Shows Photoelectron current versus Retarding potential (V) where Light intensity is constant. The graph shows curves corresponding to three different frequencies ( $v_{1} > v_{2} > v_{3}$ ), each reaching zero current at a different stopping potential ( $V_{0 (1)}$ , $V_{0 (2)}$ , $V_{0 (3)}$ ).

Conclusion from Graph: The stopping potential $V_{0}$ , and hence the maximum photoelectron energy, depends on the frequency of the light. When the retarding potential is $V_{0} = 0$ , the photoelectron current is the same for light of a given intensity regardless of its frequency.

Image/Graph: Maximum photoelectron kinetic energy $T_{ma x}$ versus frequency of incident light for three metal surfaces (Cesium, Sodium, Calcium). (The graph depicts straight lines that increase linearly, each intersecting the frequency axis at a different threshold frequency $v_{0}$ ).

Empirical Formula and Quantum Theory

Since $T = e V$ in general, $T_{ma x} = e V_{0}$ here. The relationship between $T_{ma x}$ and the frequency $v$ involves a proportionality, which can be expressed in the form:

$T_{ma x} = h (v - v_{0})$

$T_{ma x} = h v - h v_{0}$ (1)

Where $v_{0}$ is the threshold frequency below which no photoelectron occurs and $h$ is a constant. The value of $h$ is:

$h = 6.63 \times 1 0^{- 34} joule.sec$

The constant $h$ is always the same, although $v_{0}$ varies with the particular metal being illuminated.

Quantum Theory of Light

When Planck’s derivation of his formula appeared, Einstein was one of the first to understand the radical postulate of energy quantization of oscillators:

$E = nh v$ (2)

where, $n = 0, 1, 2, 3...$

A few years later, in 1905, Einstein realized that the photoelectric effect could be understood if the energy in light is not spread out over wavefronts but is concentrated in small packets, or photons. The term photon was coined by the chemist Gilbert Lewis in 1926.

Each photon of light of frequency $v$ has the energy $h v$ , the same as Planck’s quantum energy. Einstein’s break with classical physics was more drastic than Planck’s: Energy was not only given to EM waves in separate quanta but was also carried by the waves in separate quanta.

Einstein’s Interpretation

The empirical formula eqn. (1) may be rewritten:

$h v = T_{ma x} + h v_{0}$ (3)

Einstein’s proposal interprets the three terms of eqn. (3) as follows:

$h v = Energy content of each quantum of the incident light$
$T_{ma x} = Maximum photoelectron energy$
$h v_{0} = Minimum energy needed to dislodge an electron from the metal$

How Einstein’s Hypothesis Explains Observations

The three experimental observations listed above follow directly from Einstein’s hypothesis:

Because EM wave energy is concentrated in photons and not spread out, there should be no delay in the emission of photoelectrons.
All photons of frequency $v$ have the same energy, so changing the intensity of a monochromatic light beam will change the number of photoelectrons but not their energies.
The higher the frequency $v$ , the greater the photon energy $h v$ and so the more energy the photoelectrons have.

The Work Function

The minimum energy for an electron to escape from a particular metal surface is called the work function of the metal. This energy prevents electrons from pouring out all the time. The work function ( $φ$ ) is related to the threshold frequency ( $v_{0}$ ) by the formula:

Work function $φ = h v_{0}$ (3)

The greater the work function of a metal, the more energy is needed for an electron to leave its surface, and the higher the critical frequency for photoelectric emission to occur.

Table/Chart: Photoelectric Work Functions.

Metal	Symbol	Work Function, eV
Cesium	Cs	1.9
Potassium	K	2.2
Sodium	Na	2.3
Lithium	Li	2.5
Calcium	Ca	3.2
Copper	Cu	4.7
Silver	Ag	4.7
Platinum	Pt	6.4

007_index