AP Physics 2 · Unit 15 · Topic 15.5

Photon Thresholds in the Photoelectric Effect

Quantization, probability, nuclear processes, and relativity extend classical models at atomic scales.

1. Topic Lens

Photon Thresholds in the Photoelectric Effect is studied through modern physics. Connect the system boundary, interacting parts, and measurable evidence before applying a formula.

\[E=hf\]

2. Why the Formula Works

The relationship is built from definitions and conservation reasoning:

A photon carries discrete energy hf.
The material requires work function φ to release an electron.
Energy conservation leaves the remainder as the maximum electron kinetic energy.

\[K_{\text{max}}=hf-\phi\]

3. Detailed Visual Model

Photon Thresholds in the Photoelectric Effect: an original pixel-style model. Use it as a schematic, not a literal scale drawing.

4. Worked Example and Lab Link

A 5 eV photon strikes a surface with work function 2 eV. Find maximum electron kinetic energy.

Answer: Kmax=5-2=3 eV.

Investigation idea: Analyze original stopping-potential data versus frequency and infer threshold frequency and Planck-slope meaning.

Common trap: Light intensity changes photon count; below threshold frequency it does not release electrons in the ideal model.

Expanded Topic 15.5 Lesson

5. Learning Targets

Use the photon model to explain immediate electron emission and the existence of a threshold frequency.
Apply \(K_{\max}=hf-\phi\), \(eV_s=K_{\max}\), and the equivalent wavelength relationships.
Predict separately how frequency, intensity, material, and applied voltage affect photoelectric data.
Extract Planck's constant, work function, and threshold frequency from a linear graph.
Distinguish maximum electron energy from photocurrent and identify limits of the ideal one-photon model.

6. The Effect and Its Boundary Condition

In the external photoelectric effect, incident light ejects electrons from a material surface. Emission occurs only when one photon supplies at least the material's work function \(\phi\):

\[hf\ge\phi\]

The equality marks the ideal threshold. Above it, some photoelectrons leave with kinetic energy; below it, raising light intensity does not repair the energy shortage in the basic one-photon model.

7. Three Observations the Model Must Explain

No measurable buildup delayEmission begins essentially immediately even when the light is weak.

A material thresholdBelow a minimum frequency, no photocurrent appears regardless of ordinary intensity.

Energy follows frequencyMaximum photoelectron energy grows with frequency, not with intensity.

A successful model must explain all three observations at once, not merely electron emission.

8. Original Photoelectric Apparatus Model

The variable voltage separates two measurements: a forward collecting voltage reveals saturation current, while a reverse retarding voltage reveals the maximum electron kinetic energy.

9. Einstein's One-Photon Energy Ledger

One photon transfers its energy to one electron. The minimum escape cost is the work function; any remaining energy can become electron kinetic energy:

\[hf=\phi+K\]

The fastest emitted electrons require the least additional energy loss before leaving the surface, so

\[\boxed{K_{\max}=hf-\phi}\]

This maximum is an upper edge. Real photoelectrons can emerge with a distribution of smaller kinetic energies.

10. Work Function and Threshold Forms

At threshold, \(K_{\max}=0\). Therefore

\[\phi=hf_0,\qquad f_0=\frac{\phi}{h}\]

\[\lambda_0=\frac{c}{f_0}=\frac{hc}{\phi}\]

Emission by frequency\(f\ge f_0\)

Emission by wavelength\(\lambda\le\lambda_0\)

The threshold wavelength is the longest wavelength that can eject an electron, not the shortest.

11. A Fast eV–nm Toolkit

\[E_\gamma(\mathrm{eV})\approx\frac{1240}{\lambda(\mathrm{nm})}\]

If energies are in electron-volts, the numerical maximum kinetic energy in eV equals the stopping-potential magnitude in volts:

\[K_{\max}(\mathrm{eV})=V_s(\mathrm V)\]

This shortcut follows from \(1\,\mathrm{eV}=e(1\,\mathrm V)\); it is a unit relationship, not the equation \(e=1\).

12. Worked Threshold Decision

A metal has \(\phi=2.30\,\mathrm{eV}\). Its threshold wavelength is

\[\lambda_0=\frac{1240\,\mathrm{eV\,nm}}{2.30\,\mathrm{eV}}=539\,\mathrm{nm}\]

Light at 600 nm has lower photon energy and produces no photoelectrons in the ideal model. Light at 450 nm is shorter than the threshold wavelength and can eject electrons.

13. Worked Energy and Stopping-Potential Example

Light at 400 nm illuminates the same \(\phi=2.30\,\mathrm{eV}\) surface:

\[E_\gamma=\frac{1240}{400}=3.10\,\mathrm{eV}\]

\[K_{\max}=3.10-2.30=0.80\,\mathrm{eV}\]

\[V_s=\frac{K_{\max}}{e}=0.80\,\mathrm V\]

A retarding potential with magnitude 0.80 V stops even the fastest photoelectrons.

14. What Stopping Potential Actually Measures

When the collector repels electrons, each electron loses electric potential energy while crossing the tube. At the stopping condition for the fastest electrons,

\[\boxed{eV_s=K_{\max}}\]

Here \(V_s\) denotes the positive magnitude of the stopping potential. The actual applied collector voltage is retarding and may be plotted as \(-V_s\), so signs must be read from the circuit convention.

15. Original Linear Evidence Graph

For \(K_{\max}=hf-\phi\), slope is \(h\), the vertical intercept is \(-\phi\), and the horizontal intercept is \(f_0=\phi/h\). Parallel lines indicate a universal Planck constant.

16. Extract Physics from a Best-Fit Line

For a graph of \(K_{\max}\) versus \(f\), write \(y=mx+b\):

\[K_{\max}=hf-\phi\]

Slope\(h\), if energy and frequency units are consistent

Vertical intercept\(-\phi\), obtained by extrapolation

Horizontal intercept\(f_0=\phi/h\), the threshold

For a graph of \(V_s\) versus \(f\), the slope is \(h/e\) and the vertical intercept is \(-\phi/e\).

17. Example: Estimate \(h\) from Two Data Points

Suppose \(K_{\max}=0.50\,\mathrm{eV}\) at \(6.0\times10^{14}\,\mathrm{Hz}\) and \(1.33\,\mathrm{eV}\) at \(8.0\times10^{14}\,\mathrm{Hz}\). Then

\[h\approx\frac{\Delta K_{\max}}{\Delta f}=\frac{0.83\,\mathrm{eV}}{2.0\times10^{14}\,\mathrm{Hz}}=4.15\times10^{-15}\,\mathrm{eV\,s}\]

Using either point gives \(\phi\approx2.0\,\mathrm{eV}\). A many-point best-fit line is more reliable than selecting two noisy measurements.

18. Frequency and Intensity Control Different Outcomes

Change	\(K_{\max}\)	\(V_s\)	Photocurrent
Increase \(f\), fixed material and above threshold	increases	increases	not determined without photon-rate information
Increase monochromatic intensity at fixed \(f>f_0\)	unchanged	unchanged	generally increases
Increase intensity while \(f<f_0\)	no photoelectrons	not defined	remains zero in ideal one-photon model
Use a larger-\(\phi\) material at fixed \(f\)	decreases or emission stops	decreases or is absent	material-dependent

For fixed frequency, higher intensity means a larger photon arrival rate. If emission probability and collection conditions are unchanged, more electrons per second give more current.

19. Photon Flux Is Not Photon Energy

For a monochromatic beam of incident power \(P_{\mathrm{light}}\), the photon arrival rate is

\[\dot N_\gamma=\frac{P_{\mathrm{light}}}{hf}\]

At fixed \(f\), doubling power doubles photon flux but leaves each photon's energy unchanged. At fixed power, raising frequency gives more energy per photon but fewer incident photons per second. That is why current comparisons require more than frequency alone.

20. Reading Photocurrent–Voltage Curves

Retarding regionIncreasing opposition rejects progressively faster electrons until current becomes zero at \(-V_s\).

Collecting regionA positive collector attracts more emitted electrons until nearly all available electrons are collected.

Saturation currentTracks the emission/collection rate and generally grows with intensity at fixed frequency.

Same \(V_s\)Two intensity curves at the same frequency share the same maximum kinetic energy.

21. Why Electron Energies Form a Distribution

Electrons begin in different material states and depths. Some lose energy through interactions before escaping, and their paths toward the collector differ. Consequently, a photocurrent curve samples a range of kinetic energies.

The Einstein equation specifies the maximum kinetic energy for the most favorable electrons, not that every emitted electron has exactly \(hf-\phi\).

22. Classical Prediction Versus Photon Evidence

Question	Continuous classical transfer	Photon model	Observation
Weak light	long energy-accumulation delay	fewer events, but each sufficient photon acts promptly	no measurable buildup delay
Higher intensity	greater electron kinetic energy	more photons and usually more emitted electrons	current rises; \(K_{\max}\) does not
Low frequency	enough time or intensity should eventually eject electrons	each photon lacks the work-function energy	sharp material-dependent threshold

23. Experimental Limits and Extensions

The ideal equation uses a clean, specified surface; oxidation and contamination can change an effective work function.
Contact potentials and instrument calibration can shift measured voltages.
Very intense laser fields can enable multiphoton emission, which lies beyond the ordinary one-photon AP model.
In semiconductors, internal photoelectric processes and band gaps require a different device model.

Use the stated ideal assumptions unless a problem provides additional surface or material information.

24. Open Simulation Investigation

A. Find the threshold

Open PhET Photoelectric Effect.
Select one metal and keep light intensity fixed.
Scan wavelength from long to short and bracket the onset of current.
Convert the threshold wavelength to a work function and compare materials.

B. Separate energy and rate

Choose a wavelength above threshold and measure stopping potential.
Change only intensity; record current and stopping potential.
Restore intensity, change frequency, and repeat.
Write two evidence-based claims: one about photon energy and one about photon number.

25. Data-Quality Investigation

Measure \(V_s\) for at least five frequencies above threshold.
Plot \(V_s\) versus \(f\) with uncertainties and fit a straight line.
Infer \(h/e\), \(\phi/e\), and \(f_0\), including units.
Inspect residuals rather than forcing a conclusion from correlation alone.

Use a simulation or supervised low-voltage apparatus. Do not construct high-voltage or ultraviolet equipment without qualified supervision and appropriate eye protection.

26. Common Reasoning Traps

“Brighter means faster electrons.” At fixed frequency, brightness mainly changes electron rate, not \(K_{\max}\).
Reversing the wavelength test: emission requires \(\lambda\le\lambda_0\).
Giving negative kinetic energy: if \(hf-\phi<0\), report no emission in the ideal model.
Equating all electron energies with \(K_{\max}\): the measured population has a distribution.
Confusing stopping and accelerating voltages: stopping voltage opposes electron collection.
Reading the wrong graph slope: \(K_{\max}\)-versus-\(f\) has slope \(h\); \(V_s\)-versus-\(f\) has slope \(h/e\).

27. AP-Style Reasoning Checks

Light below threshold becomes ten times as intense. What changes in the ideal model?
Above threshold, frequency rises while intensity stays fixed. What happens to \(K_{\max}\)?
Above threshold, intensity doubles at fixed frequency. Compare \(V_s\) and saturation current.
What does the horizontal intercept of a \(K_{\max}\)-versus-\(f\) graph represent?
A metal has \(\phi=4.0\,\mathrm{eV}\). Can a 400 nm photon eject an electron?
Why does zero current at the stopping potential identify a maximum energy?

Answers: no photoemission begins; it increases by \(h\Delta f\); \(V_s\) stays fixed while saturation current generally doubles; threshold frequency; no, the photon has about 3.10 eV; even the fastest electrons are turned back, so all slower ones are also stopped.

Checkpoint · Topic 15.5

Explain how photon thresholds in the photoelectric effect supports or limits this conclusion: Kmax=5-2=3 eV.

Official curriculum reference: College Board AP Physics 2 course page. The explanation and worked example are independently written for this study site.