§ 33. External Photoelectric Effect: Einstein's Equation & Stopping Potential (Grade 11)
Quantum emission of photoelectrons, work function A_w, red limit λₘax, and stopping voltage Uₛtop. Light is absorbed in discrete quanta hν: the maximum kinetic energy of photoelectrons depends solely on light frequency and cathode work function.
Theory Summary & Scientific Mechanism
The external photoelectric effect, emission of electrons from a metal surface illuminated by light, defied classical wave electrodynamics, which could not explain why maximum photoelectron kinetic energy is independent of light intensity and why emission ceases abruptly above a threshold wavelength λₘax = hc / W.
In 1905, Albert Einstein resolved the paradox by proposing that light consists of discrete energy quanta (photons) of energy E_γ = hν = (hc)/(λ), where h = 6.626× 10⁻³⁴ J·s is Planck’s constant. A single electron absorbs one photon, expends the metal’s work function W to escape the surface barrier, and retains maximum kinetic energy given by Einstein’s photoelectric equation: hν = W + Eₖ,max.
Applying a reverse bias halts the photocurrent at the stopping potential Uₛ, where e Uₛ = Eₖ,max = (hc)/(λ) - W. Increasing optical intensity (for λ < λₘax) increases the photon flux and saturation photocurrent Iₛat proportionally without altering Eₖ,max.
Everyday Analogy
Solar panels, smartphone light sensors, and automatic supermarket doors all rely on the photoelectric effect! Light arrives as individual photon bullets: a flood of weak red photons cannot dislodge an electron, whereas a single energetic UV photon knocks it out instantly!
Interactive Question & Derivation
According to Einstein's photoelectric equation hν = A_w + Eₖ,max, how does doubling the light intensity I (at constant wavelength λ < λₘax) affect the maximum kinetic energy Eₖ,max of emitted photoelectrons and saturation photocurrent Iₛat?
- A: Eₖ,max and stopping voltage Uₛtop remain unchanged (determined only by photon energy hν = hc/λ), while saturation photocurrent Iₛat doubles
- B: Maximum kinetic energy Eₖ,max doubles, while photocurrent remains unchanged
- C: The red photoelectric threshold λₘax shifts toward longer wavelengths
Study Hint: By Einstein’s equation Eₖ,max = hν - W: does the maximum kinetic energy of emitted photoelectrons depend on light intensity or on light frequency ν (wavelength λ)?
Proof & Explanation: Each individual photon carries energy E_γ = hν = (hc)/(λ) ≈ (1240 eV·nm)/(λ). An electron absorbs a single photon and spends the work function A_w to escape the metal, leaving maximum kinetic energy Eₖ,max = eUₛtop = (hc)/(λ) - A_w. Increasing light intensity I increases the number of photons per second (increasing saturation current Iₛat), but does not change the energy of each photon hν!
Key Equations & 3D Simulation Parameters
E_γ = (1240)/(λ) eV, Eₖ,max = E_γ - A_w, Uₛtop = (Eₖ,max)/(e)
- Incident wavelength λ: 250 … 650 nm (default: 380 nm)
- Cathode work function A_w (Cs=1.9, K=2.2, Zn=4.3): 1.8 … 4.5 eV (default: 2.2 eV)
- Photon flux intensity Φ: 0.5 … 3 W/m² (default: 1.5 W/m²)
- Retarding voltage Uᵣet: 0 … 2.5 V (default: 0.4 V)