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F2.4 · Analyse data supporting relativity or quantum theory

Learn to analyse data supporting relativity or quantum theory through clear examples and targeted practice.

Ontario Grade 12 Physics

Revolutions in Modern Physics: Quantum Mechanics and Special Relativity

How patterns in data test models of matter, light, space, and time

In SPH3U, you used evidence such as motion measurements and wave patterns to test models. A model is a simplified description used to explain or predict observations. In this lesson, the physical system is the object or process being studied, such as light striking a metal surface or particles travelling to a detector. A reference frame is the viewpoint and coordinate system used to describe positions and times. For the examples here, choose a laboratory frame at rest with the measuring equipment. When direction matters, define the positive direction before interpreting a result. A scalar has magnitude only; a vector has magnitude and direction. Energy and time are scalars, while velocity is a vector. F2.4 asks you to analyse data that support relativity or quantum theory. Evidence can agree with a model and distinguish it from a competing explanation, but data do not prove a model in every possible situation.

What you will learn

From a question to a data pattern

Begin by identifying what was measured, the units, and which conditions were held the same. A table may show individual measurements; a graph can make a trend easier to see. Label each axis with the quantity and unit. If repeated measurements vary, look for the overall pattern rather than treating one point as decisive.
A relationship is useful when it predicts how one measured quantity changes as another changes. A straight-line trend suggests a constant rate of change. A trend that curves may suggest a different relationship or that another factor matters. A graph supports a model when the model predicts the observed pattern and competing models do not explain it as well.

Quantum evidence: light and electrons

In the photoelectric effect, light shines on a metal and electrons leave its surface. The frequency of light is the number of wave cycles passing a point each second, measured in hertz. The stopping potential is the voltage needed to prevent the fastest emitted electrons from reaching a collector. Its magnitude can be used to find their maximum kinetic energy.
The photon model treats light energy as arriving in packets called photons. For a given metal, the model predicts a minimum frequency, called the threshold frequency. Below that frequency, increasing the light intensity does not release electrons. Above it, the maximum electron energy increases with frequency. This frequency pattern supports the photon model; intensity mainly affects how many electrons are emitted.
Electrons also produce diffraction patterns when directed at a suitable crystal. Diffraction is the spreading and reinforcement pattern associated with waves. A particle model that treats electrons only as tiny classical objects does not account for the observed pattern. The pattern supports the quantum description of electrons, while not making electrons identical to ordinary water or sound waves.
Ek,max⁡=hf−ϕ=eVsE_{k,\max}=hf-\phi=eV_s

Relativity evidence: time measurements

Special relativity relates measurements made in different inertial reference frames. An inertial frame is one moving at constant velocity relative to another, with no acceleration. A central prediction is time dilation: a clock moving relative to the laboratory is measured to run more slowly than a clock at rest in the laboratory.
Cosmic-ray muons provide evidence. Muons are short-lived particles produced high in Earth’s atmosphere. Detectors at the surface record more muons than a simple calculation using the muons’ short rest-frame lifetime and their travel distance would predict. In the laboratory frame, the muons move at high speed and their measured lifetime is longer. This pattern agrees with time dilation.
The comparison must use the same reference frame consistently. In the laboratory frame, use the muon’s speed and the distance through the atmosphere to find travel time. Compare that time with the muon lifetime measured in that frame. A data set or report should be identified as measured evidence; a classroom calculation using supplied values is an analysis, not a new measurement.
Δt=γΔt0,γ=11−v2/c2\Delta t=\gamma\Delta t_0,\qquad \gamma=\frac{1}{\sqrt{1-v^2/c^2}}

A reliable analysis routine

First define the system and frame. Next name the measured quantities and their SI units. Then state the relationship predicted by the model before substituting values. If the data are in a table, compare points or calculate a slope where appropriate. A slope is the change in the vertical quantity divided by the change in the horizontal quantity.
Finally, interpret the result. Check that units match the quantity, that a sign or direction agrees with your convention, and that the size is physically reasonable. Explain how the observed pattern supports the model. Do not claim that a single data set rules out every alternative explanation.

Worked example

Finding Planck’s constant from photoelectric data

A sample photoelectric data set for one metal gives stopping potentials of 0.40 V at 6.0 × 10¹⁴ Hz and 1.23 V at 8.0 × 10¹⁴ Hz. Use the change in the data to estimate Planck’s constant. Treat these as supplied example data, not as measurements made in this lesson.
  1. Define the measurements
    The system is light striking the metal and the emitted electrons. The reference frame is the laboratory frame, and no direction is needed for the scalar stopping potential. The unknown is the slope relating stopping potential to frequency.
  2. Use the model
    The photoelectric relationship predicts that the slope of stopping potential versus frequency is Planck’s constant divided by the elementary charge. Rearranging gives the constant from the measured changes.
    h=eΔVsΔfh=e\frac{\Delta V_s}{\Delta f}
  3. Substitute and calculate
    Use the change between the two data points. The voltage change is in volts and the frequency change is in hertz. Use the course value e=1.60×10−19 Ce=1.60\times10^{-19}\,\mathrm{C}.
    h=(1.60×10−19 C)1.23 V−0.40 V(8.0−6.0)×1014 Hz=6.6×10−34 J sh=(1.60\times10^{-19}\,\mathrm{C})\frac{1.23\,\mathrm{V}-0.40\,\mathrm{V}}{(8.0-6.0)\times10^{14}\,\mathrm{Hz}}=6.6\times10^{-34}\,\mathrm{J\,s}
Answer: The estimated value is 6.6×10−34 J s6.6\times10^{-34}\,\mathrm{J\,s}.
Check: Since 1 V=1 J/C1\,\mathrm{V}=1\,\mathrm{J/C}, the units reduce to joule-seconds. The estimate is close to the accepted value, about 6.63×10−34 J s6.63\times10^{-34}\,\mathrm{J\,s}. Two significant figures match the supplied data.

Worked example

Checking a photon-energy pattern

In a supplied data set, light at 5.0×1014 Hz5.0\times10^{14}\,\mathrm{Hz} produces no photoelectrons, while light at 7.0×1014 Hz7.0\times10^{14}\,\mathrm{Hz} does. The measured work function of the metal is 3.0×10−19 J3.0\times10^{-19}\,\mathrm{J}. Find the model’s threshold frequency and compare it with the observations.
  1. Define the system
    The system is the metal surface and incident light in the laboratory frame. Frequency and energy are scalars. The unknown is the threshold frequency, the minimum frequency predicted to eject electrons.
  2. Relate threshold to work function
    At threshold, the photon energy just matches the energy needed to release an electron. There is no kinetic energy left over at this minimum.
    hf0=ϕhf_0=\phi
  3. Calculate and compare
    Divide the work function by Planck’s constant. Then compare the result with the two frequencies in the supplied data.
    f0=3.0×10−19 J6.63×10−34 J s=4.5×1014 Hzf_0=\frac{3.0\times10^{-19}\,\mathrm{J}}{6.63\times10^{-34}\,\mathrm{J\,s}}=4.5\times10^{14}\,\mathrm{Hz}
Answer: The predicted threshold frequency is 4.5×1014 Hz4.5\times10^{14}\,\mathrm{Hz}. The observed no-emission frequency is below it, while the emission frequency is above it.
Check: Joules cancel, leaving inverse seconds, or hertz. The ordering of the observations matches the threshold prediction and supports the photon-energy model.

Worked example

Testing time dilation with muons

For a simplified supplied example, a muon travels at 0.98c0.98c. Its rest-frame lifetime is 2.2 μs2.2\,\mathrm{\mu s}. Find its lifetime as measured in the laboratory frame. Do not treat these values as newly measured in this lesson.
  1. Set the frame and direction
    The system is one muon. The reference frame is the laboratory, and choose the muon’s direction of travel as positive. Its velocity is a vector in that direction. The lifetime is a scalar.
  2. Find the time factor
    Use the special-relativity time-dilation relationship. The ratio v/cv/c is dimensionless, so the factor has no units.
    γ=11−(0.98c)2/c2=5.0\gamma=\frac{1}{\sqrt{1-(0.98c)^2/c^2}}=5.0
  3. Calculate laboratory lifetime
    Multiply the rest-frame lifetime by the time factor. Round to two significant figures, consistent with the supplied speed.
    Δt=5.0(2.2 μs)=11 μs\Delta t=5.0(2.2\,\mathrm{\mu s})=11\,\mathrm{\mu s}
Answer: The laboratory-frame lifetime is approximately 11 μs11\,\mathrm{\mu s}.
Check: The units remain microseconds, and the result is longer than the rest-frame lifetime, as the model predicts. Actual cosmic-muon evidence comes from detector observations; this calculation illustrates how supplied values are analysed.

Common mistakes and how to avoid them

Saying that brighter light must give photoelectrons more energy.
Correction: In the photon model, frequency determines energy per photon. Greater intensity can increase the number of photons and often the number of emitted electrons.
Treating a model calculation as a laboratory measurement.
Correction: Label supplied, historical, or simulated values accurately. A calculation using them is an analysis, not a new experiment.
Comparing lifetimes without stating the reference frame.
Correction: Name the frame for each time measurement and use the time-dilation relationship consistently.
Claiming that one matching pattern proves a theory completely.
Correction: Say that the evidence supports the model because its predictions agree with the observed pattern.

Lesson summary

Check your understanding

Question 1

For one metal, which observation best supports the photon model in the photoelectric effect?
  1. Below a threshold frequency, increasing intensity still produces electrons.
  2. Below a threshold frequency, increased intensity does not produce electrons, while higher-frequency light does.
  3. Stopping potential depends only on the distance from the lamp.
  4. correctIndex'': 1,
Show answer and explanation
Below a threshold frequency, increased intensity does not produce electrons, while higher-frequency light does.
A threshold frequency is predicted by the photon model. Increasing intensity below the threshold does not give individual photons enough energy to release electrons.

Question 2

A graph of stopping potential against frequency is a straight line. What does its slope represent according to the photoelectric model?
  1. Planck’s constant divided by the elementary charge.
  2. The metal’s work function multiplied by frequency.
  3. The number of emitted electrons per second.
  4. correctIndex'': 0,
Show answer and explanation
Planck’s constant divided by the elementary charge.
The photon model predicts a linear relationship whose stopping-potential slope is h/eh/e.

Question 3

In the laboratory frame, a muon’s speed is close to the speed of light. What does time dilation predict about its measured lifetime compared with its rest-frame lifetime?
  1. It is shorter in the laboratory frame.
  2. It is the same in every frame.
  3. It is longer in the laboratory frame.
  4. correctIndex'': 2,
Show answer and explanation
It is longer in the laboratory frame.
The laboratory-frame interval is the rest-frame interval multiplied by a factor greater than one for nonzero speed.

Key terms

Data
Recorded observations or measurements, including their units and conditions.
Model
A simplified description used to explain observations and make predictions.
Photon
A packet of light energy in the photon model.
Threshold frequency
The minimum light frequency predicted to eject electrons from a particular metal.
Reference frame
A viewpoint and coordinate system used to describe measurements.
Time dilation
The predicted difference in measured time intervals for clocks in relative motion.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation F2.4. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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