Questions: Planck-Einstein Relation: Energy and Frequency

Question 1

You shine two light beams on a metal with a work function of 2.5 eV. Beam A is dim violet light (λ ≈ 400 nm, ~3.1 eV per photon). Beam B is intense red light (λ ≈ 700 nm, ~1.8 eV per photon). Which beam ejects electrons?

• A. Beam B, because its greater intensity delivers more total energy to the metal surface
• B. Beam A only, because each violet photon's energy exceeds the work function threshold, regardless of the beam's intensity
• C. Both beams together, because their combined energy exceeds the threshold
• D. Neither beam — continuous wave energy delivery is required to eject electrons

Answer: (B)

Whether a photon can eject an electron depends entirely on whether its individual energy hf exceeds the work function — a threshold that must be crossed by a single photon in a single interaction. Red photons each carry 1.8 eV, below the 2.5 eV threshold; no number of red photons overcomes this because the energy is not cumulative. Dim violet photons each carry 3.1 eV, exceeding the threshold, so even a small number eject electrons immediately. Intensity controls how many electrons are ejected per second, not whether ejection occurs at all.

Question 2

Doubling the intensity of a green laser beam while keeping its wavelength fixed:

• A. Doubles the energy of each photon
• B. Doubles the frequency of the light
• C. Doubles the number of photons per second while leaving each photon's individual energy unchanged
• D. Doubles both the number of photons and their individual energies

Answer: (C)

Intensity (power per unit area) scales with the number of photons per second, not with individual photon energy. Each photon's energy is fixed by its frequency through E = hf; changing intensity means changing the photon flux, not the frequency. This is the quantization insight: the energy per quantum is set by the frequency alone, and more intense light means more quanta, not more energetic quanta.

Question 3

A beam of bright red light has photons with more energy per photon than a beam of dim blue light.

• T. True
• F. False

Answer: False

Photon energy depends on frequency (E = hf), not on the intensity of the beam. Blue photons have higher frequency than red photons and therefore more energy per photon — regardless of how many there are. Brightness (intensity) measures photons per second per unit area; it has no bearing on the energy carried by each individual photon. This is the core misconception E = hf is designed to correct: energy per photon is a property of the frequency, not the beam.

Question 4

The fact that radio waves cannot ionize atoms while X-rays can is a direct consequence of the Planck-Einstein relation E = hf and the large difference in frequency between the two types of radiation.

• T. True
• F. False

Answer: True

Radio waves have frequencies around 10⁸ Hz, giving photon energies of roughly 10⁻⁶ eV — far below atomic binding energies (~10 eV). X-rays have frequencies around 10¹⁸ Hz, giving photon energies of ~10 keV — well above atomic binding energies. The E = hf relation directly maps the electromagnetic spectrum onto a scale of photon energies that determines whether radiation interacts with atomic electrons (ionizing) or passes through matter without exciting them (non-ionizing).

Question 5

Why can a single ultraviolet photon eject an electron from a metal surface when a million radio photons cannot, and what does this reveal about the nature of electromagnetic energy?

Model answer: Because the photoelectric interaction is a one-photon, one-electron event: a single photon must carry enough energy to overcome the metal's work function in one interaction. Radio photons each carry energy far below this threshold (E = hf is tiny for radio frequencies), and energy from multiple photons doesn't accumulate to eject a single electron. A UV photon's frequency is high enough that hf exceeds the threshold, so ejection occurs immediately. This reveals that electromagnetic energy is quantized: it is exchanged in discrete packets whose size depends only on frequency, not on intensity.

This insight — that electromagnetic energy comes in indivisible quanta of size hf — is what Einstein's 1905 paper established and what the photoelectric effect proves experimentally. The classical picture would allow low-intensity light to slowly deliver enough energy to eject electrons; the quantum picture says no, because delivery is always in one-photon chunks. The quantum of energy sets a hard threshold: either the chunk is big enough or it isn't.