Theory — Photons, Work Function, and the Threshold
- Discuss the principles and applications of the photoelectric effect.
- Solve problems relating photon energy, work function, and electron ejection.
- Determine the work function of a metal from kinetic-energy-versus-frequency data.
Light as Photons
In the photoelectric effect, light striking a metal surface ejects electrons. The surprising result, explained by Einstein in 1905, is that light delivers its energy in discrete packets called photons. The energy of a single photon depends only on its frequency f (or equivalently its wavelength λ), not on how bright the light is.
h = 6.626 × 10⁻³⁴ J·s (Planck's constant)
c = 3.00 × 10⁸ m/s
Convenient form: E (eV) = 1240 / λ (nm)
Work Function and the Threshold Frequency
Each metal binds its electrons with a minimum energy called the work function ϕ. A photon must carry at least this much energy to free an electron. If the photon energy is below ϕ, no electron is ejected no matter how intense the light. The frequency at which a photon's energy exactly equals ϕ is the threshold frequency f₀; the matching wavelength is the threshold wavelength λ₀.
Threshold frequency: f₀ = ϕ / h
Threshold wavelength: λ₀ = h·c / ϕ = 1240 / ϕ(eV) nm
Einstein's Photoelectric Equation
When a photon with more than enough energy is absorbed, the excess appears as kinetic energy of the ejected electron. The most energetic electrons carry the full surplus:
Measured with a stopping voltage: e·V_stop = KE_max
So V_stop = (h·f − ϕ) / e (and KE_max in eV equals V_stop in volts)
Reading the KE-vs-Frequency Graph
Einstein's equation has the form of a straight line, y = mx + b, with KE_max on the y-axis and frequency on the x-axis. The slope of that line is Planck's constant h, the y-intercept is −ϕ (so the work function is read off directly), and the x-intercept is the threshold frequency f₀. Remarkably, the slope is the same for every metal — only the intercept shifts — which is exactly what the photon model predicts.
Raise the frequency
Above threshold, higher frequency means more energetic photons, so KE_max and the stopping voltage increase. The slope of KE vs f stays fixed at h.
Raise the intensity
Brighter light at the same frequency sends more photons, so more electrons are ejected — but each electron's maximum kinetic energy is unchanged. (Intensity is the subject of Part 2.)
| Metal | Work function ϕ (eV) | Threshold λ₀ (nm) | Threshold f₀ (Hz) |
|---|---|---|---|
| Sodium (Na) | 2.28 | 544 | 5.51 × 10¹⁴ |
| Calcium (Ca) | 2.87 | 432 | 6.94 × 10¹⁴ |
| Zinc (Zn) | 4.30 | 288 | 1.04 × 10¹⁵ |
| Copper (Cu) | 4.70 | 264 | 1.14 × 10¹⁵ |
Apparatus
The equipment used to eject electrons with light and measure their maximum kinetic energy. In the simulation these are modelled for you, but the readings correspond to what each instrument would measure.
Instructions — Running the Virtual Experiment
Use the built-in simulation to investigate how the frequency of light affects electron emission, and to determine the work function of four metals. Record every reading in your lab notebook; include your KE-versus-frequency graph and calculations in your report.
Simulation — Photoelectric Apparatus & KE-vs-Frequency Plot
Controls
Metal
Set wavelength, then calculate KE_max
From your graph (2+ points)
| λ (nm) | f (×10¹⁴ Hz) | Your KE_max (eV) | Measured KE_max (eV) |
|---|---|---|---|
| No readings yet — pick a wavelength above threshold, calculate KE_max, and click "Record & check". | |||
Team Questions
Example Lab Report
Sample report demonstrating the expected format and level of detail. Use as a guide for your own submission.
Photoelectric Effect — Determining the Work Functions of Four Metals
Physics | Section: [Your Section] | Date: [Date]
Lab Members: [Names of all members present]
Purpose
To investigate how the frequency of incident light controls the kinetic energy of photoelectrons, to determine the work function of four metals (sodium, calcium, zinc, and copper) from kinetic-energy-versus-frequency graphs, and to verify that the slope of each graph yields Planck's constant.
Theory
A photon of frequency f carries energy E = hf. When it is absorbed by an electron in a metal, the electron escapes only if the photon energy exceeds the work function ϕ; the surplus becomes kinetic energy. Einstein's photoelectric equation has the form of a straight line in f.
slope m = h · y-intercept = −ϕ · x-intercept f₀ = ϕ/h
E(eV) = 1240 / λ(nm) · V_stop = KE_max / e
Plotting KE_max against f therefore gives a line whose slope is Planck's constant (the same for every metal) and whose intercept reveals the work function of the particular metal.
Calculations — Sample: Sodium (ϕ = 2.28 eV) at λ = 400 nm
Photon energy: E = 1240/400 = 3.10 eV
Maximum kinetic energy: KE_max = 3.10 − 2.28 = 0.82 eV
Stopping voltage: V_stop = KE_max/e = 0.82 V
Slope from two points (300 nm and 400 nm): m = (1.853 − 0.820) eV / (1.000 − 0.750)×10¹⁵ Hz = 4.13 × 10⁻¹⁵ eV·s. Converting: h = m·e = (4.13 × 10⁻¹⁵)(1.602 × 10⁻¹⁹) = 6.62 × 10⁻³⁴ J·s ✓
Results Table — Sodium KE_max vs Frequency
| λ (nm) | f (×10¹⁴ Hz) | E = hf (eV) | KE_max (eV) | V_stop (V) |
|---|---|---|---|---|
| 300 | 10.00 | 4.13 | 1.85 | 1.85 |
| 350 | 8.57 | 3.54 | 1.26 | 1.26 |
| 400 | 7.50 | 3.10 | 0.82 | 0.82 |
| 450 | 6.67 | 2.76 | 0.48 | 0.48 |
| 500 | 6.00 | 2.48 | 0.20 | 0.20 |
| Metal | Measured ϕ (eV) | Accepted ϕ (eV) | Measured slope h (J·s) |
|---|---|---|---|
| Sodium | 2.28 | 2.28 | 6.62 × 10⁻³⁴ |
| Calcium | 2.87 | 2.87 | 6.62 × 10⁻³⁴ |
| Zinc | 4.30 | 4.30 | 6.62 × 10⁻³⁴ |
| Copper | 4.70 | 4.70 | 6.62 × 10⁻³⁴ |
Discussion
For every metal the maximum kinetic energy rose linearly with frequency, exactly as KE_max = hf − ϕ predicts. The slope of each line was 6.62 × 10⁻³⁴ J·s, agreeing with the accepted value of Planck's constant (6.626 × 10⁻³⁴ J·s) and — crucially — coming out the same regardless of which metal was used. This is the signature of the photon model: the energy carried per photon depends only on frequency, not on the target.
The y-intercept of each line gave the work function directly: 2.28 eV for sodium, 2.87 eV for calcium, 4.30 eV for zinc, and 4.70 eV for copper, all matching their accepted values. Below each metal's threshold frequency no electrons were emitted, and increasing the light intensity below threshold made no difference — confirming that a single photon, not the accumulated brightness, must supply the escape energy. This behaviour cannot be explained by treating light purely as a wave.
Conclusion
The experiment confirmed Einstein's photoelectric equation. Maximum kinetic energy was linear in frequency with slope equal to Planck's constant, the threshold frequency marked the onset of emission, and the work functions extracted from the intercepts matched accepted values for all four metals. The results support the particle (photon) model of light.
Practice Questions
Show all work and include units in your answers.