Einstein’s Photoelectric Equation & the Energy Quantum of Radiation ⚡

1 · Why the old wave picture fell short 🔍

The classic wave model said electrons soak up light energy little by little across the entire wavefront. So, turning up brightness (intensity) should crank up each electron’s energy and a high-enough intensity, whatever the color (frequency), ought to knock electrons out without any special “cut-off” color. Experiments disagree: kinetic energy does not rise with intensity, and there is a sharp threshold frequency. Even worse, calculations show an electron would need hours to collect enough energy, yet emission actually happens instantly! :contentReference[oaicite:0]{index=0}

2 · Einstein’s fresh idea (1905) 💡

Light arrives in tiny packets called photons, each carrying energy \(h\nu\). An electron that grabs one photon either (a) pops out immediately or (b) stays put—no slow “filling.” If the photon’s energy beats the metal’s work function \(\phi_0\), the electron shoots out with  \(K_{\text{max}} = h\nu – \phi_0\)  (Eq. 11.2). :contentReference[oaicite:1]{index=1}

  • Linear link: \(K_{\text{max}}\) climbs straight-line with frequency, untouched by intensity.
  • One-photon game: every emission is a single photon-single electron handshake, so no delay ⏱️.
  • Brightness: stronger light just brings more photons ⇒ more electrons, not faster ones.

3 · Threshold frequency 🚦

Emission only starts when \(h\nu > \phi_0\) or \[ \nu_0 = \frac{\phi_0}{h} \] (Eq. 11.3). Below \(\nu_0\) the metal stays quiet, whatever the intensity. :contentReference[oaicite:2]{index=2}

4 · Stopping potential 🛑

To halt the quickest photoelectrons we apply a reverse voltage \(V_0\). Einstein’s equation rewrites as \[ eV_0 = h\nu – \phi_0 \quad (\nu \ge \nu_0) \] —a straight-line plot of \(V_0\) versus \(\nu\) whose slope gives \(h/e\). :contentReference[oaicite:3]{index=3}

5 · Millikan’s precision tests 🧪

Between 1906-1916, R. A. Millikan measured that slope for many alkali metals and nailed the same Planck constant \(h\) found in entirely different experiments—confirming, rather than refuting, Einstein’s formula. :contentReference[oaicite:4]{index=4}

6 · Photon snapshot 🌟

  • Energy  \(E = h\nu = \dfrac{hc}{\lambda}\)
  • Momentum  \(p = \dfrac{h\nu}{c} = \dfrac{h}{\lambda}\) :contentReference[oaicite:5]{index=5}
  • Speed  \(c\) (always!), electrically neutral 🚫⚡
  • Intensity ↑ ⇒ more photons per second, each with the same energy.
  • Energy & momentum stay conserved in photon-particle collisions; photon count may change.

7 · Important NEET take-aways 🎯

  1. Einstein’s equation \(K_{\text{max}} = h\nu – \phi_0\) (straight-line graph with slope \(h\)).
  2. Threshold frequency formula \(\nu_0 = \phi_0/h\); no emission below it.
  3. Stopping-potential relation \(eV_0 = h\nu – \phi_0\).
  4. Electron energy depends on light frequency, while emitted current depends on intensity.
  5. Photon momentum \(p = h/\lambda\) often shows up in Compton and related problems.