22.2 Photoelectric Effect Evidence and Einstein's Equation

In the classic tube experiment, light falls on a metal photosurface and emitted electrons are collected by a second plate. By biasing the collector negatively, only the fastest electrons can reach it. Increase this retarding voltage until current falls to zero: that magnitude is the stopping voltage, and it directly reports the maximum emitted kinetic energy through eV_s.

Two observations immediately challenge a pure-wave energy-transfer picture: first, increasing intensity at fixed frequency raises current but does not raise stopping voltage. Second, below a threshold frequency no electrons are emitted, however intense the light is. These results are difficult to reconcile with gradual energy accumulation from a continuous wave at a single electron.

This equation explains all four core observations at once. Higher frequency means larger hf and therefore larger maximum kinetic energy. Intensity at fixed frequency raises the number of photons striking per second, so more electrons are emitted per second (larger current) while each electron still receives the same per-event photon energy. Threshold emerges naturally when hf = phi.

Graphically, plotting E_K,max versus frequency gives a straight line with slope h and horizontal intercept f_c = phi/h. Different metals produce parallel lines (same slope h) with different intercepts (different work functions). In stopping-voltage form, V_s = (h/e)f - phi/e, so slope is h/e and intercept is material-dependent.

Another frequent error is mixing units: if you use h in SI units, energies must be in joules. If you use the shortcut hc ≈ 1240 eV·nm, stay in eV and nm consistently. A final check is sign convention for stopping voltage: equations often use magnitude V_s while circuit diagrams may show negative collector potential relative to emitter.