It is observed that light of 430 nm just begins to emit photoelectrons. What is the quantum prediction if the wavelength of the incident light is changed to 400 nm?

Question

BlackTom AI · Accepted Answer

Photon energy E = hc/λ, so shortening the wavelength raises the energy per photon.

1) Relevant concepts
- Photoelectric effect: electrons are emitted only if photon energy E ≥ work function φ. The threshold wavelength λth satisfies φ = hc/λth. Photocurrent (number of electrons per second collected) is proportional to the number of photoelectrons emitted per second; when the photon energy exceeds φ each absorbed photon can liberate at most one electron (assuming unity quantum efficiency for the argument), so more energetic photons produce electrons immediately and with higher kinetic energy.

2) Step‑by‑step reasoning
- Given: 430 nm “just begins” to emit photoelectrons → 430 nm is effectively the threshold wavelength λth, so φ = hc/(430 nm).
- Change to 400 nm: compute photon energies to show the 400 nm photons exceed the threshold.
  - Use hc = 6.626×10^-34 J·s × 3.00×10^8 m/s = 1.9878×10^-25 J·m.
  - E(430 nm) = 1.9878×10^-25 / 430×10^-9 = 4.62×10^-19 J ≈ 2.88 eV (threshold).
  - E(400 nm) = 1.9878×10^-25 / 400×10^-9 = 4.97×10^-19 J ≈ 3.10 eV.
- Since E(400 nm) > φ, 400 nm photons will eject electrons with excess kinetic energy K = E(400) − φ ≈ 0.22 eV. Therefore emission is immediate and stronger than at the threshold condition where emission was just beginning.

3) Why the correct answer is correct
- At the threshold wavelength emission is minimal; moving to a shorter wavelength immediately produces more energetic photons that can liberate electrons more readily and produce a larger photocurrent (assuming the same incident photon flux or the same light source conditions). The quantum prediction is an immediate increase in photocurrent when the wavelength is reduced below threshold.

4) Why other choices are incorrect
- “Increases gradually” contradicts the quantum prediction that emission turns on as soon as E > φ (an immediate change). “Decreases” options are wrong because higher photon energy cannot reduce the number of emitted electrons in this basic photoelectric picture. “No change” is wrong because at threshold there was barely emission; going below threshold produces a clear increase. “Cannot be determined” is unnecessary because the threshold information suffices.

Conclusion: The photocurrent increases immediately, matching answer a.

It is observed that light of 430 nm just begins to emit photoelectrons. What is the quantum prediction if the wavelength of the incident light is changed to 400 nm?单项选择题

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