Photoelectric apparatus and stopping potential

Light shining on a metal knocks electrons out of it. In this model, brighter light releases more of them each second but none faster, while light of a higher frequency makes the fastest ones faster, as Einstein expected if light gives up its energy in separate quanta.

Section 8 applies the light quantum to cathode rays produced by light. In the simplest picture, which Einstein says he will assume, one quantum gives its whole energy hν to one electron; the electron spends a work Φ, characteristic of the body, in leaving it, so the fastest electrons come out with kinetic energy hν − Φ. A body charged positive just enough to keep them all in, to the potential V_s, satisfies eV_s = hν − Φ. Two consequences follow. Plotted against ν, V_s is a straight line whose slope, h/e, does not depend on the substance. And if each quantum acts independently of the rest, the intensity of the light changes how many electrons leave each second, not how fast they go. The defaults, 600 THz light at 1 mW on a hypothetical surface with Φ = 2.2 eV, give quanta of 2.48 eV, a threshold at 532 THz and a stopping potential of 0.281 V; with one quantum in ten releasing an electron, 2.52 × 10¹⁴ electrons leave each second, a current of 40.3 μA. Doubling the power doubles the current and leaves 0.281 V unchanged.

Start with one quantum. Its energy is hν, Planck's constant times the frequency: h = 6.626 × 10⁻³⁴ J s and ν = 6.00 × 10¹⁴ Hz give hν = 3.976 × 10⁻¹⁹ J. An electronvolt, the energy an electron gains in falling through one volt, is 1.602 × 10⁻¹⁹ J, so hν = 3.976/1.602 = 2.48 eV. To leave the metal an electron must do the work Φ, here 2.2 eV. If the quantum gives all its energy to one electron at the surface, the most that electron can keep is 2.481 − 2.2 = 0.281 eV. To stop it, charge the metal positive: an electron climbing back against a potential V loses eV of energy, so it turns back once eV equals its kinetic energy, and the fastest one needs V_s = 0.281 V. That is the stopping potential. A quantum that carries less than Φ releases nothing, so the light must have hν at least Φ, or ν at least ν₀ = Φ/h = (2.2 × 1.602 × 10⁻¹⁹)/(6.626 × 10⁻³⁴) = 5.32 × 10¹⁴ Hz, which is 532 THz. Now the intensity. A power of 1 mW delivers 10⁻³ J each second; shared into quanta of 3.976 × 10⁻¹⁹ J, that is 10⁻³/(3.976 × 10⁻¹⁹) = 2.52 × 10¹⁵ quanta each second. If one in ten releases an electron, 2.52 × 10¹⁴ electrons leave each second, and since each carries 1.602 × 10⁻¹⁹ C the current is 2.52 × 10¹⁴ × 1.602 × 10⁻¹⁹ = 4.03 × 10⁻⁵ A, or 40.3 μA. Double the power to 2 mW and there are twice as many quanta, so twice the current, 80.6 μA; each quantum still carries 2.48 eV, so the fastest electron still needs 0.281 V to stop. Einstein's own check sets the work to nothing, Φ = 0, and takes ν = 1.03 × 10¹⁵ Hz, where the solar spectrum ends in the ultraviolet. Then eV_s = hν gives 4.26 V with today's constants. He printed 4.3 V and said it agrees in order of magnitude with Lenard's results.

/**
 * Stopping potential magnitude: V_s = K_max / e = (h * nu - Phi) / e (in Volts).
 *
 * Epistemic rule:
 * When nu < nu_0, stopping potential is `not-applicable` with reason
 * "no emitted electron in this model".
 * Under partial transfer, returns `underdetermined` with upperBound = (h * nu - Phi) / e.
 */
export function stoppingPotentialMagnitude(
  nu: number,
  workFunctionJoules: number,
  set?: ConstantSet,
  transferModelOrOptions:
    | EnergyTransferMode
    | Readonly<{ transferModel?: EnergyTransferMode }> = "complete",
): PhotoelectricResult<number> {
  const transferModel: EnergyTransferMode =
    typeof transferModelOrOptions === "object" && transferModelOrOptions !== null
      ? (transferModelOrOptions.transferModel ?? "complete")
      : transferModelOrOptions;
  const kResult = kMax(nu, workFunctionJoules, set, transferModel);
  if (kResult.status === "underdetermined") {
    const e = getElementaryCharge(set);
    const extra: {
      upperBound?: number;
      unit?: string;
      citation?: string;
      reason?: string;
      bounds?: readonly [number, number];
      quantityId?: string;
    } = {
      unit: "V",
      quantityId: "stoppingPotentialMagnitude",
    };
    if (kResult.upperBound !== undefined) extra.upperBound = kResult.upperBound / e;
    if (kResult.citation !== undefined) extra.citation = kResult.citation;
    if (kResult.reason !== undefined) extra.reason = kResult.reason;
    if (kResult.bounds !== undefined) extra.bounds = [kResult.bounds[0] / e, kResult.bounds[1] / e];
    return underdetermined(kResult.compatibleFamily, kResult.neededInformation, extra);
  }
  if (kResult.status !== "value") return kResult;
  const e = getElementaryCharge(set);
  return ok(kResult.value / e, "V", {
    ...(kResult.note ? { note: kResult.note } : {}),
    quantityId: "stoppingPotentialMagnitude",
  });
}