Einstein's Photoelectric Explanation Revision Notes | A Level AQA Physics

1. The Crisis That Changed Physics

Picture this: physicists in the late 19th century had a beautifully complete theory of light. Maxwell's equations described electromagnetic waves with stunning precision. Then experimenters shone light onto metal surfaces and watched electrons fly off — and the results made absolutely no sense under wave theory. This was not a minor discrepancy. It was a full-blown crisis.

The photoelectric effect — the ejection of electrons from a metal surface by incident electromagnetic radiation — forced physicists to abandon centuries of thinking about the nature of light. Einstein's 1905 explanation of this effect is one of the most significant moments in the history of physics, and it is the reason he won the Nobel Prize, not relativity.

2. What Classical Wave Theory Predicted — and Why It Failed

Before Einstein, the accepted view was that light is a continuous electromagnetic wave. Wave theory made very specific predictions about what should happen when light hits a metal surface. Every single one of those predictions turned out to be wrong.

Here is the core idea from wave theory: energy is spread continuously across the wavefront. A more intense beam carries more energy per second. A lower-frequency (dimmer, redder) beam carries less energy per second, but given enough time, it should still deliver enough energy to free an electron. That was the prediction. That is not what happens.

Each row in that table is a separate nail in the coffin of wave theory. The existence of a sharp threshold frequency is particularly damning: wave theory has no mechanism to produce a cut-off. A wave of any frequency, given enough time, should accumulate enough energy at the surface. Experiment says otherwise, unambiguously.

These failures are not edge cases — they are systematic and reproducible. Classical wave theory is structurally incapable of explaining them. A fundamentally new model of electromagnetic radiation was needed.

3. Einstein's Photon Model: The Explanation

In 1905, Einstein proposed a radical idea: electromagnetic radiation is not a continuous wave but is instead quantised into discrete packets of energy called photons. Each photon carries a fixed amount of energy determined solely by the frequency of the radiation.

The crucial shift is this: a photon interacts with one electron in a one-to-one collision. The photon either has enough energy to free that electron, or it does not. There is no accumulation of energy from multiple photons — each interaction is all-or-nothing. This single idea resolves every failure of wave theory at a stroke.

When a photon strikes the metal surface, it gives all of its energy to a single electron. Some of that energy is used to overcome the binding energy holding the electron in the metal — this is the work function, $\phi$. Any energy left over becomes the kinetic energy of the emitted electron.

The word maximum in $E_{k(\text{max})}$ matters. Electrons deeper in the metal lose additional energy escaping to the surface, so they emerge with less kinetic energy. The equation gives the upper limit — the kinetic energy of an electron that escapes from the very surface with no additional energy loss.

3a. How the Photon Model Resolves Each Failure

• Threshold frequency: A photon must have energy $hf \geq \phi$ to free an electron. If $f < f_0$, no single photon has enough energy, so no emission occurs — regardless of how many photons arrive (i.e. regardless of intensity).
• Instantaneous emission: The photon-electron interaction is immediate. There is no waiting for energy to build up. One photon, one electron, one interaction.
• Intensity increases number of electrons (not their energy): A brighter source sends more photons per second. More photons means more one-to-one interactions, so more electrons are emitted. But each photon still has the same energy $hf$, so $E_{k(\text{max})}$ is unchanged.
• Frequency increases $E_{k(\text{max})}$: Higher frequency means higher photon energy $hf$. With $\phi$ fixed for a given metal, more energy is left over as kinetic energy: $E_{k(\text{max})} = hf - \phi$. This is a direct linear relationship.

4. Threshold Frequency and the Work Function

At the threshold frequency $f_0$, the photon has just enough energy to free an electron — but none left over for kinetic energy. Setting $E_{k(\text{max})} = 0$ in Einstein's equation gives the relationship between threshold frequency and work function.

Different metals have different work functions because the strength of the electrostatic binding of surface electrons varies. Caesium has a low work function (around 2.0 eV) and responds to visible light. Zinc has a higher work function and requires ultraviolet radiation. This is why not all metals show the photoelectric effect under visible light.

Energy in photoelectric calculations is often quoted in electron-volts (eV). One electron-volt is the energy gained by an electron accelerated through a potential difference of 1 V: $1 \text{ eV} = 1.6 \times 10^{-19}$ J. Always convert to joules before substituting into $hf = \phi + E_{k(\text{max})}$.

5. Worked Calculations Using the Photoelectric Equation

The photoelectric equation is straightforward to apply, but exam questions often require you to rearrange it, convert units, or extract $f_0$ from a graph. Work through the following examples carefully — the method is transferable to any photoelectric calculation.

Worked example: Finding Maximum Kinetic Energy from Frequency and Work Function

Ultraviolet radiation of frequency $1.50 \times 10^{15}$ Hz is incident on a metal surface with a work function of $3.60 \times 10^{-19}$ J. Calculate the maximum kinetic energy of the emitted electrons. ($h = 6.63 \times 10^{-34}$ J s)

1. $hf = \phi + E_{k(\text{max})}$ — This is the starting point for all photoelectric calculations. [M1]
2. $E_{k(\text{max})} = hf - \phi$ — Subtract the work function from the photon energy to find the kinetic energy left over.
3. $hf = 6.63 \times 10^{-34} \times 1.50 \times 10^{15}$ $hf = 9.945 \times 10^{-19}$ J — Multiply the Planck constant by the frequency. Keep full precision at this stage. [M1]
4. $E_{k(\text{max})} = 9.945 \times 10^{-19} - 3.60 \times 10^{-19}$ $E_{k(\text{max})} = 6.345 \times 10^{-19}$ J — Subtract the work function. Both values are in joules, so no unit conversion is needed here. [A1]
5. $E_{k(\text{max})} \approx 6.35 \times 10^{-19}$ J — The data was given to 3 significant figures, so round the answer to 3 significant figures.

Answer: $E_{k(\text{max})} = 6.35 \times 10^{-19}$ J

Worked example: Calculating Threshold Frequency from Work Function

The work function of sodium is $3.78 \times 10^{-19}$ J. Calculate the threshold frequency for sodium. ($h = 6.63 \times 10^{-34}$ J s)

1. $\phi = hf_0$ — At the threshold frequency, all photon energy goes into overcoming the work function — none is left as kinetic energy. [M1]
2. $f_0 = \frac{\phi}{h}$ — Divide both sides by the Planck constant.
3. $f_0 = \frac{3.78 \times 10^{-19}}{6.63 \times 10^{-34}}$ — Both values carry their units: J divided by J s gives Hz (s$^{-1}$). [M1]
4. $f_0 = \frac{3.78}{6.63} \times 10^{-19-(-34)} = 0.5701... \times 10^{15}$ $f_0 = 5.70 \times 10^{14}$ Hz — Divide the coefficients: $3.78 \div 6.63 = 0.5701...$. Then handle the powers: $10^{-19} \div 10^{-34} = 10^{-19+34} = 10^{15}$. [A1]

Answer: $f_0 = 5.70 \times 10^{14}$ Hz

6. Graphical Analysis: $E_{k(\text{max})}$ Against Frequency

One of the most powerful pieces of evidence for Einstein's model comes from plotting the maximum kinetic energy of emitted electrons against the frequency of incident radiation. The result is a straight line — and every feature of that line carries physical meaning.

Rearranging Einstein's equation as $E_{k(\text{max})} = hf - \phi$ puts it in the form $y = mx + c$. The gradient of the line is the Planck constant $h$, the x-intercept is the threshold frequency $f_0$, and the y-intercept (extrapolated) is $-\phi$. This graph is a direct experimental verification of the photon model.

The line does not begin at the origin — it starts at $f_0$ on the frequency axis, where $E_{k(\text{max})} = 0$. For $f < f_0$, there are no data points because no emission occurs. The gradient is the same for all metals (it equals $h$), but the x-intercept shifts depending on the work function of the metal being used.

7. Significance: Wave-Particle Duality of Electromagnetic Radiation

Einstein's explanation of the photoelectric effect did far more than solve a single experimental puzzle. It established that electromagnetic radiation has a dual nature: it behaves as a wave in phenomena such as interference and diffraction, and it behaves as a stream of discrete photons in phenomena such as the photoelectric effect.

This was a profound conceptual shift. Before 1905, the wave model of light was considered essentially complete. Einstein's work showed that the wave model, while correct in many contexts, is not the whole picture. Electromagnetic radiation is neither purely a wave nor purely a particle — it is something more subtle that exhibits both characters depending on the experimental context.

The significance extends beyond light. Einstein's photon concept, building on Planck's earlier quantisation of energy, was a founding contribution to quantum mechanics. It demonstrated that quantisation is not a mathematical convenience but a physical reality — energy is exchanged in discrete packets at the atomic scale. This insight underpins the entire framework of modern physics.

8. Exam Technique and Common Mistakes

Photoelectric effect questions appear regularly in AQA A Level papers and tend to target the same conceptual traps. The following guidance covers the most mark-losing errors and the language that secures full credit.

For extended-response questions asking you to evaluate Einstein's explanation, structure your answer as: (1) state what classical wave theory predicted, (2) state what was actually observed, (3) explain how Einstein's photon model accounts for the observation, and (4) comment on the broader significance for the nature of electromagnetic radiation. This structure maps directly onto the mark scheme's progression from recall through analysis to evaluation.

Einstein's photoelectric explanation stands as the definitive demonstration that electromagnetic radiation is quantised — a result that classical wave theory is fundamentally unable to produce, and that reshaped the entire trajectory of 20th-century physics.