Einstein Photoelectric Equation

11.5 Wave Theory Cannot Explain the Photoelectric Effect

If light were a continuous electromagnetic wave (as Maxwell's theory says), the energy delivered to the metal would scale with intensity. Three predictions follow:

1. Brighter light should give photoelectrons of higher kinetic energy — more energy soaks into each electron.
2. The effect should occur at any frequency, provided the beam is intense enough.
3. For a faint beam, the electron should accumulate energy slowly; a measurable time-lag of seconds, even minutes, should appear before emission begins.

Every one of these predictions fails when measured. The maximum kinetic energy depends on frequency only; below \(\nu_0\) no current flows however bright the beam; and the emission, even at the lowest detectable intensities, is essentially instantaneous (\(<10^{-9}\) s).

11.6 Einstein's Photoelectric Equation — Energy Quantum of Radiation

In 1905, working from Planck's 1900 hypothesis that radiation is emitted in discrete packets of energy \(h\nu\), Albert Einstein proposed a radically simple resolution: light itself is granular. A monochromatic beam of frequency \(\nu\) is a stream of energy quanta, each of energy \(h\nu\). When such a quantum is absorbed by an electron in the metal, the entire energy \(h\nu\) is transferred in one indivisible event.

Of this energy, the electron must spend at least \(\phi_0\) (the work function) to escape the surface. Whatever remains shows up as kinetic energy of the freed photoelectron. The most loosely bound electrons (those at the very surface) lose only \(\phi_0\); deeper-lying electrons lose more, so they emerge with smaller kinetic energy. Hence:

Einstein's photoelectric equation (Eq 11.2)

Three immediate consequences follow naturally:

• Threshold frequency: Photoelectrons can be liberated only if \(h\nu \geq \phi_0\), i.e. \(\nu \geq \nu_0\) where \[\nu_0 = \frac{\phi_0}{h}\]
• Linear KE–frequency relation: \(K_\text{max}\) varies linearly with \(\nu\); the slope is the same for every metal — the universal constant \(h\).
• Intensity ↔ photon number: A more intense beam carries more photons per second but each still has energy \(h\nu\). So intensity sets the saturation current, not \(K_\text{max}\).

11.7 Stopping Potential and Work Function

Multiplying the equation by 1 and using \(K_\text{max} = eV_0\) (where \(V_0\) is the stopping potential measured in the experiment):

This is exactly the linear graph observed by Hallwachs and Lenard. A plot of \(V_0\) versus \(\nu\) gives:

• A straight line of slope \(h/e\) — the same for every metal.
• An x-intercept at \(\nu_0 = \phi_0/h\) — different for each metal.
• A y-intercept at \(-\phi_0/e\) — directly giving the work function.

11.8 Millikan's Verification (1916)

Robert A. Millikan, originally sceptical of Einstein's quantum hypothesis, performed precise measurements of \(V_0\) versus \(\nu\) for several alkali metals. His results were a perfect straight line with the predicted slope \(h/e\). Once he inserted the known electronic charge \(e\) (which he himself had measured in the famous oil-drop experiment), the value of Planck's constant emerged:

This agreement, both quantitative and structural, settled the case in favour of Einstein's quantum picture and led to the 1921 Nobel Prize for the photoelectric law (and the 1923 Nobel Prize for Millikan).

Worked Examples