Reasoning Chain: Why $V_{0}$ is Independent of Intensity

Step 1 — What is stopping potential? $V_{0}$ is the minimum retarding voltage applied to the collector that reduces photocurrent to exactly zero. At V = −$V_{0}$, even the fastest photoelectron (with KE_max) is stopped.

Step 2 — Energy of the fastest photoelectron The fastest electron is one that escapes directly from the metal surface with no internal collisions. By energy conservation: energy absorbed from photon = work function + kinetic energy → hν = φ + KE_max → KE_max = hν − φ.

Step 3 — Connecting KE_max to $V_{0}$ Work done by retarding field on electron: W = e$V_{0}$ (force × distance = charge × potential). At stopping: e$V_{0}$ = KE_max = hν − φ. Therefore: $V_{0}$ = (hν − φ)/e.

Step 4 — Where is intensity in this equation? The equation $V_{0}$ = (hν − φ)/e contains only h (constant), ν (frequency of light), φ (property of metal), and e (fundamental charge). Intensity does NOT appear anywhere.

Step 5 — Physical interpretation of intensity Intensity = (number of photons per second) × (energy per photon) = N × hν. If intensity doubles (N → 2N), twice as many electrons are ejected per second → photocurrent doubles. But each individual photon still carries energy hν — the energy absorbed per electron is unchanged → KE_max unchanged → $V_{0}$ unchanged.

Step 6 — Conclusion $V_{0}$ is determined by the energy balance for a single photon-electron interaction. Intensity controls how many such interactions occur per second (current), but not the energy of each interaction (KE per electron). Therefore, $V_{0}$ is completely independent of intensity and depends only on frequency.

Reasoning chain summary: ν determines energy per photon → energy per electron → KE_max → $V_{0}$. Intensity determines number of photons → number of electrons → photocurrent. These are parallel, independent processes.