Einstein's Photoelectric Equation

Einstein's photoelectric equation expresses energy conservation for a single photon-electron interaction: the photon energy $h\nu$ splits into the work function ${\varphi }_0$ needed to free the electron and the leftover kinetic energy, giving $K_{max}=h\nu -{\varphi }_0=\frac{hc}{\lambda }-{\varphi }_0$. The photon energy for $400\text{nm}$ light is $\frac{1240}{400}=3.1\text{eV}$ using $hc=1240\text{eV nm}$. Subtracting the work function, $K_{max}=3.1-2.0=1.1\text{eV}$. The option $3.1\text{eV}$ is just the photon energy and forgets to subtract ${\varphi }_0$. The option $5.1\text{eV}$ wrongly adds the work function instead of subtracting it. The option $0.9\text{eV}$ uses an incorrect photon energy. Because the calculated value is positive, emission does indeed occur, confirming the incident photons exceed the threshold. This single-photon energy balance, which classical wave theory could not reproduce, earned Einstein the Nobel Prize and is the heart of the dual-nature chapter. A consistency check shows $K_{max}<h\nu$, exactly as the conservation statement demands. It is important to recognise that increasing the intensity of this 400 nm beam would liberate more electrons per second but would not change the 1.1 eV maximum energy of any single one, because that energy is fixed by frequency alone. The complete failure of the classical wave picture to predict this frequency dependence, together with the instantaneous nature of emission, is precisely what forced the adoption of the photon model.