Photoelectric Effect and Wave Theory of Light

The Failure of the Wave Theory

By the end of the nineteenth century, the wave nature of light was firmly established. Phenomena like interference, diffraction, and polarisation were explained beautifully by treating light as an electromagnetic wave — a travelling disturbance of electric and magnetic fields, with energy spread continuously across the wavefront. This picture was so successful that it seemed natural to ask: can it also explain the photoelectric effect?

The answer, as we shall see, is a clear no.

To understand why, we need to examine what the wave picture predicts for photoelectric emission. According to this view, when light falls on a metal surface, the free electrons at the surface absorb energy continuously from the wave. The intensity of the radiation determines the amplitude of the electric and magnetic fields — higher intensity means larger amplitude, which means more energy is available per unit area per unit time.

Here is what the wave theory would then predict:

• Prediction 1: The greater the intensity of light, the more energy each electron absorbs per second. Therefore, the maximum kinetic energy of the emitted photoelectrons should increase with increasing intensity.
• Prediction 2: Since energy is absorbed continuously, given enough time, even a low-frequency but sufficiently intense beam should be able to transfer enough energy to an electron to overcome the work function. There should be no threshold frequency — any frequency of light, if intense enough, should eventually cause emission.
• Prediction 3: The absorption of energy happens across the entire wavefront, shared among a very large number of electrons. The energy absorbed per electron per unit time is therefore tiny. Explicit calculations show that it would take hours or even longer for a single electron to accumulate enough energy to escape the metal.

Now compare these predictions with the actual observations from the previous section:

• Observation (i): The maximum kinetic energy of photoelectrons is independent of intensity, and depends only on frequency.
• Observation (ii): There is a threshold frequency below which no emission occurs, no matter how intense the light.
• Observation (iii): For frequencies above the threshold, the photoelectric current is proportional to the intensity.
• Observation (iv): The emission is instantaneous — there is no measurable time lag.

The wave picture of light, for all its successes in explaining interference and diffraction, is fundamentally unable to account for the most basic features of photoelectric emission. A radically new idea was needed.

Einstein's Revolutionary Proposal: Light Quanta

In 1905, Albert Einstein proposed a bold new picture of electromagnetic radiation to resolve this crisis. His central idea was this: radiation energy is not a continuous wave, but is built up of discrete units — quanta of energy. Each quantum of radiant energy has an energy given by

where $h$ is Planck's constant ($6.626 \times 10^{-34} \text{ J s}$) and $\nu$ is the frequency of the light. These quanta are what we now call photons.

In Einstein's picture, the photoelectric effect is not a gradual accumulation of energy from a wave. Instead, it is a single, elementary process: one electron absorbs one quantum of radiation (one photon) in a single instant. If the energy of that photon, $h\nu$, is greater than the minimum energy needed for the electron to escape the metal surface (the work function $\phi_0$), the electron is emitted. The excess energy appears as the kinetic energy of the emitted electron.

This leads directly to Einstein's photoelectric equation:

Here, $K_{\text{max}}$ is the maximum kinetic energy of the emitted photoelectrons. Electrons that are more tightly bound (deeper inside the metal) will emerge with less kinetic energy, but the maximum possible value is given by this equation.

How Einstein's Equation Explains All Observations

Einstein's simple equation accounts for every one of the puzzling observations in a clean, elegant way.

1. Dependence of Kinetic Energy on Frequency (and Independence from Intensity)

From $K_{\text{max}} = h\nu - \phi_0$, we see that $K_{\text{max}}$ depends linearly on the frequency $\nu$. It is completely independent of the intensity of the radiation. This matches observation (i) perfectly.

Why? Because the photoelectric effect is a single-photon, single-electron process. Each electron absorbs the energy of exactly one photon. The intensity tells you how many photons are arriving, but each photon's energy is fixed by its frequency. A more intense beam does not give more energy to any individual electron — it just allows more electrons to participate.

2. Existence of a Threshold Frequency

Since kinetic energy cannot be negative, the equation $K_{\text{max}} = h\nu - \phi_0$ implies that photoelectric emission is possible only if

or, equivalently,

This $\nu_0$ is the threshold frequency. For any frequency below $\nu_0$, the photon energy $h\nu$ is less than the work function, so no electron can gain enough energy to escape — no matter how intense the beam (i.e., no matter how many such low-energy photons arrive). This explains observation (ii).

3. Photoelectric Current is Proportional to Intensity

For frequencies above the threshold ($\nu > \nu_0$), the number of photoelectrons emitted per second (the photoelectric current) is proportional to the intensity of the incident radiation. Why? Because intensity is proportional to the number of photons per unit area per unit time. More photons mean more electrons can absorb a photon and be emitted. This explains observation (iii).

4. Instantaneous Emission

The elementary process is the absorption of a single photon by a single electron. This is an instantaneous event — there is no gradual build-up of energy. Whether the beam is intense (many photons) or weak (few photons), the basic act of absorption takes no time at all. A weak beam does not cause a delay; it simply means fewer electrons are emitted per second. This explains observation (iv).

The Stopping Potential and a Crucial Prediction

Recall that the maximum kinetic energy of photoelectrons can be measured by applying a stopping potential $V_0$ that just brings them to rest:

where $e$ is the electronic charge. Substituting this into Einstein's equation gives:

or

This is a powerful result. It predicts that if you plot the stopping potential $V_0$ against the frequency $\nu$ of the incident light, you will get a straight line.

• The slope of this line is $\displaystyle \frac{h}{e}$, a universal constant that is independent of the material of the metal.
• The intercept on the $\nu$-axis gives the threshold frequency $\nu_0$.
• The intercept on the $V_0$-axis gives $-\phi_0/e$. …