# Photoelectric Effect and The Quantum Model of Light

This is part of the HSC Physics course under the topic Light: Quantum Model.

### HSC Physics Syllabus

• investigate the evidence from photoelectric effect investigations that demonstrate inconsistency with the wave model of light (ACSPH087, ACSPH123, ACSPH137)
• analyse the photoelectric effect Kmax = hf – φ as it occurs in metallic elements by applying the Law of Conservation of Energy and the photon model of light.

### Hertz’s observation of photoelectric effect

The photoelectric effect was first observed (but not explained) in 1887 by Heinrich Hertz and is therefore sometimes termed the Hertz effect. He discovered that shining UV light on a receiver loop increased the spark length.

Maxwell's electromagnetic model of light underpinned the classical explanation of the photoelectric effect, where the electrons in the metal atoms were shaken and caused to vibrate by the oscillating electric field of the light. Eventually some of these electrons would be shaken loose and ejected. The ejected electrons are called photoelectrons.

It is important to understand how this classical theory predicted the way in which the number and speed of photoelectrons emitted, along with the time required for their emission, would vary with the intensity and wavelength of the incident light.

### Predictions made by the Wave Model of Light

The wave model of light made 3 main predictions regarding the photoelectric effect:

• Light with higher intensity will eject electrons with greater kinetic energy
• Light with higher frequency will result in a greater current
• Light of any intensity and frequency will be able to eject electrons

In the wave model, intensity of light is proportional to the square of its amplitude. Increasing the intensity of light increases the extent of oscillation of electric fields. Therefore, electrons will be ejected with greater kinetic energy when light of higher intensity is used to illuminate a metal surface.

Frequency of light is related to the rate of oscillation of light's electric field. When light of higher frequency is used, the rate at which electrons will be ejected is faster. This will result in a greater current.

$$Intensity = \frac{P}{A}$$

In the wave model, energy is assumed to be continuous. Electrons in the metal were ejected when they receive a sufficient amount of energy. When light of higher intensity (greater power) is used, electrons will take shorter time to receive this sufficient amount of energy and be ejected. Therefore, the wave model of light predicted that electrons will be ejected from a metal regardless of what the intensity or frequency of the light source is.

### Lenard's Experiment on the Photoelectric Effect

In 1902, Hertz's student, Philipp Lenard, studied how the energy of the emitted photoelectrons varied with the intensity and frequency of light. He also measured the magnitude of current produced by the ejected electrons.

The variable light source was used to illuminate a positively charged metal plate (cathode) in an evacuated tube (vacuum). The ejected electrons, which are termed photoelectrons, were travelling towards the negatively charged metal (anode). Measuring number of photoelectrons

When photoelectrons reach the anode, they flow through the wire from the anode back to the cathode. The galvanometer is used to measure the current, which is the amount of charge (electrons) flowing through it per second. When there are more photoelectrons reaching the anode, the magnitude of current will increases.

Measuring the kinetic energy of photoelectrons

To measure the kinetic energy of the ejected electrons, Lenard applied a reverse voltage that does work against the liberated electrons' motion. Thus, only electrons ejected with enough kinetic energy to overcome the work done by the electric field will reach the anode and flow through the galvanometer.

The stopping voltage is the potential difference required to prevent photoelectrons from reaching the anode. The stopping voltage can be used to calculate the maximum kinetic energy of photoelectrons.

$$K_{max} = q_eV_{stopping}$$

When photoelectrons have more kinetic energy, the stopping voltage is higher.

Lenard's experimental observations were inconsistent with predictions made by the wave model of light

• Lenard observed that increasing intensity of light did not increase the stopping voltage (a measure of photoelectrons' kinetic energy). Instead, the photocurrent increased. However, the classical predicted that BOTH photocurrent and stopping voltage would increase.
• Increasing frequency did not increase the photocurrent but instead increased the stopping voltage.
• Light below a certain frequency was unable to liberate photoelectrons, no matter what the intensity of light was. There existed a threshold frequency - a minimum frequency required for photoelectrons to be ejected. This was inconsistent with the wave model, which predicted that the energy barrier for electrons to be ejected can simply be met by increasing the intensity of light.

### Einstein’s Quantum Model of Light

Key Takaways

• Quantum model of light: light consists of photons.
• When a photon strikes a metal surface, it transfers its energy completely to one electron. The energy transfer only occurs when the photon's energy exceeds the metal's work function.
• Remaining amount of energy is transformed into a photoelectron's kinetic energy.
• If a photon's energy is less than the work function, no photoelectrons are emitted.

In 1905, Einstein developed the quantum model of light. The quantum model of light describes light as consisting of photons which are discrete packets of energy (mirroring Planck's quanta). Einstein borrowed Planck's quantum theory and proposed that the energy of a photon is given by

$$E = hf$$

Einstein proposed that when the incoming light is incident on an object, a complete energy transfer occurs strictly from one photon to one electron. That is, no electron can absorb the energy from more than one photon, and no photon can transfer its energy to more than one electron.

It is important to note that Einstein's idea of quantisation differed from that of Planck's. Planck theorised that there were 'oscillators' of some sort in atoms that vibrated at discrete frequencies and thus emitted discrete energies according to E = hf. In this way,

Planck still perceived light as a continuous, unbroken wave. In contrast, Einstein's quantum model of light describes light as discrete packets of energy (photons).

What is Work Function?

In the photoelectric effect, work function (φ) is the minimum amount of energy required to remove an electron from the metal. Think of this as the electrostatic cost required to remove electrons from the metal's lattice structure.

Einstein explained that energy transfer between a photon and an electron only occurs if the photon's energy exceeds the metal's work function. Since the energy of a photon depends on its frequency, a threshold frequency is required for photons to eject electrons.

To eject photoelectrons, the electrostatic attractions within the metal must be overcome. As such, there must be a minimum amount of energy required.

The work function is related to Lenard's observation of the 'threshold frequency' (f0) mentioned above.

These two quantities can be related by:

$$\phi = hf_0$$

Since the strength of electrostatic attractions varies among metals, the work function and threshold frequency will differ among metals.

### Law of Conservation of Energy in Photoelectric Effect

After the work function of the metal is overcome, the remaining energy of a photon is transformed into a photoelectron's kinetic energy.

The law of conservation of energy implies that the energy required to liberate the electrons (work function) plus that in excess (kinetic energy) must equal that which was absorbed (energy of photon).

Therefore,

$$K_{max} = hf - \phi$$

The stopping voltage required to stop photoelectrons from reaching the anode is given by

$$q_eV_{stopping} = hf - \phi$$

Photoelectrons vary in kinetic energy

If the electron is some distance into the material of the cathode (away from the surface), some energy will be lost as it moves towards the surface. This means the final kinetic energy possess by electrons ejected from deep down in the cathode will be lower. The most energetic electrons (greatest kinetic energy) emitted will be those very close to the surface.

### How does Einstein’s quantum model of light explain the photoelectric effect?

In Einstein's quantum model, light's intensity is proportional to the number of photons. Thus, an increase in intensity corresponds to an increase in the number of photons emitted by the light source per second.

Lenard's observation #1: light below a certain frequency cannot eject electrons regardless of its intensity • A photon can only transfer its energy to an electron if its energy exceeds the metal's work function.
• If light's frequency is below the threshold frequency, no electrons are ejected.
• Intensity of light (number of photons) does not change the energy of one photon. An electron can only absorb energy from one photon only.

Lenard's observation #2: stopping voltage increases with the frequency of light, not intensity

• The stopping voltage is dependent on the maximum kinetic energy of photoelectrons.
• When a photon transfers its energy to an electron, the excess is transformed into its kinetic energy.
• The kinetic energy of photoelectrons increases with photon's energy. Thus, the stopping voltage increases with photon's frequency.
• Intensity does not affect the stopping voltage because the energy of a photon does not depend on light's intensity.

Lenard's observation #3: current increases with the intensity of light, no frequency • Intensity of light is proportional to the number of photons.
• If a photon's energy exceeds the work function, more photons incident on the metal will cause more photoelectrons to be emitted. When more photoelectrons are emitted and reach the anode, a greater current is observed.
• Frequency does not affect the magnitude of current because the energy of a photon does not determine the number of photoelectrons that can be emitted as it cannot be transferred to more than one electron.

### Graphs in Photoelectric Effect Experiments

Maximum Kinetic Energy vs Frequency

Typically, the maximum kinetic energy of photoelectrons can be plotted against the frequency of light used to irradiate the metal Note the following features of the graph:

• The gradient of the line of best fit represents Planck’s constant h.
• The y-intercept represents the negative value of work function. This can be experimentally determined by extrapolating the line of best fit.
• The x-intercept represents the threshold frequency of light required to produce the photoelectric effect.
• When a different metal is used, the gradient of the line of best fit should remain unchanged as it represents Planck's constant. The x and y-intercepts will change.

Current vs Frequency • Below a certain frequency, no photoelectrons will be emitted because a photon does not have sufficient energy (less than the work function).
• When the frequency is above the threshold frequency, current is independent of frequency because number of photoelectrons emitted is unaffected by the energy (frequency) of a photon.

Maximum Kinetic Energy vs Intensity

The maximum kinetic energy of photoelectrons is independent of the light's intensity. • Photoelectrons' maximum kinetic energy depends on the energy of a single photon, which is determined by its frequency.
• Intensity of light (number of photons) does not affect the kinetic energy because an electron cannot absorb energy from more than one photon.

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Next section: Theory of Special Relativity: Einstein's Postulates