Why experimental physics: measurement as the test of theory

Every equation in a physics textbook earned its place because someone measured something. Newton's law of gravitation was believed because it predicted the orbits Tycho Brahe had recorded and Kepler had summarized; Maxwell's equations were believed after Heinrich Hertz generated and detected radio waves in 1887 and 1888; quantum mechanics was believed because it predicted spectra, specific heats and scattering patterns to many decimal places. Theory proposes and experiment disposes. This course is about the second half of that sentence: how physicists design experiments, build and understand their apparatus, analyse their data honestly, attach believable uncertainties to their results, and communicate what they found. The previous courses taught you what is known. This one teaches you how it came to be known, and how you can add to it.

Experimental physics is a craft as well as a science. It rewards patience, scepticism toward one's own results, careful records and a habit of asking what could be going wrong. It also rewards cleverness: many of the great experiments, from Michelson's interferometer to the lock-in amplifier that pulls a nanovolt signal out of noise, are triumphs of design rather than of brute force. Over twelve lessons we will build the tools you need, then apply them to a set of classic experiments that every physicist should understand from the inside: Millikan's oil drops, Thomson's charge-to-mass ratio, the photoelectric effect, the Franck–Hertz experiment and optical interferometry. The capstone is about writing it all up.

What a measurement is

A measurement compares an unknown quantity with a standard. When you read 0.800 m on a metre stick, you are counting how many times a defined unit fits along the pendulum string; when a digital voltmeter displays 4.762 V, it has compared the input with an internal voltage reference through a chain of calibrations that leads, ultimately, to the definitions of the International System of Units. Since 2019 every SI base unit has been defined by fixing the numerical value of a constant of nature: the second through the caesium hyperfine frequency, the metre through the speed of light, the kilogram through Planck's constant, the ampere through the elementary charge, the kelvin through Boltzmann's constant. Several of those constants were themselves measured, over a century, by experiments you will meet in this course.

A measurement is never a single number. It is a best estimate together with a statement of how far the true value might reasonably be from it. Writing \(g = 9.81\ \text{m/s}^2\) says almost nothing; writing \(g = 9.807 \pm 0.012\ \text{m/s}^2\) says that the experimenter believes the true value lies, with roughly two chances in three, between 9.795 and 9.819. Without the uncertainty, a result cannot be compared with a theory, with another measurement or with an accepted value, so it cannot test anything. The central skill of this course is producing honest uncertainties.

Accuracy, precision and the two kinds of error

Two words are often confused. Precision describes how closely repeated measurements agree with each other; accuracy describes how close they are to the true value. A result can be precise without being accurate if every measurement shares the same error. The deviations from the true value come in two kinds. Random errors vary unpredictably from one trial to the next: the reaction time of a hand on a stopwatch, the thermal noise in a resistor, the jitter of a reading on a scale. They average down as you repeat the measurement, roughly as one over the square root of the number of trials. Systematic errors shift every trial the same way: a stretched tape measure, a clock that runs slow, a formula that neglects an effect present in the apparatus. They do not average down, and repetition cannot reveal them. Finding systematic errors requires thought, cross-checks and, often, an entirely different method.

A first experiment: the pendulum

The simple pendulum is the traditional first experiment, and it already contains everything this course is about. For small swings, a mass on a string of length \(L\) has period

Measure \(L\) and \(T\) and you have \(g\). But the formula is an approximation. It assumes a point mass on a massless string, no air drag and, above all, small amplitude. The exact period at amplitude \(\theta_0\) is longer: it can be computed exactly from the arithmetic-geometric mean, \(T = T_0/\mathrm{AGM}(1, \cos(\theta_0/2))\), and for small amplitudes it is approximately \(T_0(1 + \theta_0^2/16)\). At 10 degrees the correction is about 2 parts in 1000. That sounds small, but it is a systematic error: it biases every measurement of \(g\) the same way, and if your timing is good enough it will dominate your result.

The program below simulates a pendulum of length 0.800 m swinging at 10 degrees in a laboratory where \(g = 9.81\ \text{m/s}^2\). A student times it with a hand stopwatch. Starting and stopping a stopwatch by hand introduces a random error of about a tenth of a second, mostly reaction time. The student repeats each measurement ten times, timing either a single swing, ten swings or fifty swings, and computes \(g\) with the small-angle formula.

import math, random
random.seed(6)
# A simple pendulum timed with a hand stopwatch. True values: L = 0.800 m, g = 9.81 m/s^2.
L, g = 0.800, 9.81
T0 = 2*math.pi*math.sqrt(L/g)                        # small-angle period
def period(theta0):                                    # exact period via the arithmetic-geometric mean
    a, b = 1.0, math.cos(theta0/2)
    while abs(a - b) > 1e-15: a, b = (a + b)/2, math.sqrt(a*b)
    return T0/a
amp = math.radians(10)
T = period(amp)
print(f"small-angle period {T0:.4f} s, true period at 10 degrees {T:.4f} s ({(T/T0 - 1)*1e3:.2f} parts per thousand longer)")
# Each stopwatch reading has a random reaction error of 0.10 s (start and stop combined).
def time_swings(n, trials=10):
    vals = [(n*T + random.gauss(0, 0.10))/n for _ in range(trials)]
    m = sum(vals)/trials
    s = math.sqrt(sum((v - m)**2 for v in vals)/(trials - 1))
    return m, s, s/math.sqrt(trials)
for n in (1, 10, 50):
    m, s, se = time_swings(n)
    gm = 4*math.pi**2*L/m**2
    dg = gm*2*se/m
    print(f"timing {n:2d} swings x 10 trials: T = {m:.4f} +/- {se:.4f} s  ->  g = {gm:.3f} +/- {dg:.3f} m/s^2")
# The finite amplitude is a systematic effect: ignoring it biases g low.
gb = 4*math.pi**2*L/T**2
print(f"g from the exact 10-degree period with the small-angle formula: {gb:.4f} m/s^2 (bias {gb - g:+.4f})")

small-angle period 1.7943 s, true period at 10 degrees 1.7977 s (1.91 parts per thousand longer)
timing  1 swings x 10 trials: T = 1.7794 +/- 0.0360 s  ->  g = 9.975 +/- 0.404 m/s^2
timing 10 swings x 10 trials: T = 1.7974 +/- 0.0031 s  ->  g = 9.776 +/- 0.034 m/s^2
timing 50 swings x 10 trials: T = 1.7980 +/- 0.0006 s  ->  g = 9.769 +/- 0.006 m/s^2
g from the exact 10-degree period with the small-angle formula: 9.7727 m/s^2 (bias -0.0373)

Three lessons are already visible. First, timing one swing is nearly useless: the reaction error of 0.10 s is more than 5 parts in 100 of the period, and even after ten repeats the result for \(g\) is uncertain by 0.4 m/s\(^2\). Timing many swings divides the same reaction error by the number of swings, so ten swings is ten times better and fifty swings fifty times better. Good experimental design attacks the largest source of uncertainty directly, and here the cheap fix of counting more swings beats any amount of repetition.

Second, as the random error shrinks, the systematic one appears. With fifty swings the result is \(9.769 \pm 0.006\ \text{m/s}^2\), which is many standard errors below the true 9.81. Nothing is wrong with the arithmetic or the stopwatch: the small-angle formula, applied to the real 10-degree period, gives 9.773, and the measurement agrees with that biased value. A careless experimenter might report the precise but wrong number, or might conclude that local gravity is anomalous. A careful one recognizes that the quoted uncertainty covers only the random scatter, asks what assumptions the formula makes, and either corrects for amplitude or reduces it.

Third, the statistical uncertainty itself is an estimate. With ten trials, the standard deviation of the sample is known only to about 25 parts in 100, which is why the one-swing result can land somewhat closer to or further from the truth than its quoted error suggests. Lessons 3 and 4 make these ideas quantitative.

Experiments that changed physics

It is worth recalling how often a single careful measurement has overturned theory. In 1887 Albert Michelson and Edward Morley set out to measure the motion of the Earth through the luminiferous ether with an interferometer sensitive to a fraction of a fringe. They expected a shift of about four tenths of a fringe as the apparatus rotated; they saw less than a hundredth. That null result, repeated and refined, became one of the experimental foundations of special relativity. In 1909 Hans Geiger and Ernest Marsden, working under Ernest Rutherford, found that a few alpha particles bounced back from thin gold foil, which the then-favoured model of the atom could not explain; Rutherford's nuclear atom followed in 1911. In 1964 Arno Penzias and Robert Wilson could not get rid of a faint, uniform microwave hiss in their horn antenna, even after cleaning out pigeon droppings; it was the cosmic microwave background, the afterglow of the hot early universe.

The modern era continues the pattern. The Sudbury Neutrino Observatory, two kilometres underground in a nickel mine in Ontario, showed in 2001 and 2002 that neutrinos from the Sun change flavour on the way, solving the solar neutrino problem and earning Arthur McDonald a share of the 2015 Nobel Prize; its successor laboratory, SNOLAB, hosts dark matter searches today. In 2015 the LIGO detectors measured a change in arm length of less than a thousandth of a proton's diameter and detected gravitational waves from merging black holes. Each of these results depended not only on a good idea but on understanding the apparatus so well that a surprising number could be trusted.

The same history also shows the opposite danger. In 2011 the OPERA collaboration reported neutrinos apparently travelling faster than light from CERN to the Gran Sasso laboratory in Italy. The collaboration itself treated the result with suspicion and invited scrutiny; within months the anomaly was traced to a loose fibre-optic connector and a clock oscillator problem, and the corrected measurement agreed with relativity. In 2014 the BICEP2 team announced a pattern in the polarization of the microwave background that they interpreted as a signature of cosmic inflation; joint analysis with Planck satellite data showed the signal was largely due to dust in our own galaxy. Neither episode was fraud; both were reminders that extraordinary claims need the most rigorous search for mundane explanations.

The experimental cycle

Most experiments follow a cycle. You begin with a question, often prompted by theory: does the period really scale as \(\sqrt{L}\)? Is charge quantized? You design an experiment that is sensitive to the answer and insensitive to everything else, estimate what precision you will need and what errors might creep in, then build or assemble the apparatus. A pilot run usually shows that something does not work as expected, and you redesign. Then you take data, recording everything, including what seemed unimportant at the time. You analyse the data, propagate uncertainties, test your model against the data with tools such as chi-squared, and look hard for systematic effects. Finally you write up the result so that someone else could reproduce it, and you expose it to criticism. The cycle is iterative: analysis often sends you back to the bench.

Throughout this course you will see a theme: the experimenter's job is not to get the right answer. It is to find out what the answer is, and to say how well it is known. Sometimes that means reporting a result that disagrees with the textbook, after checking it as carefully as possible. Sometimes it means admitting that the experiment cannot distinguish between two theories. The integrity of physics rests on experimenters reporting what they actually found.

Where physics is measured

Experimental physics happens at every scale. In teaching laboratories, students repeat the classic experiments with apparatus that has changed little in a century. In university research groups, a few people build tabletop experiments in atomic physics, condensed matter or optics. At national facilities, hundreds or thousands collaborate on accelerators and detectors. In Canada these include TRIUMF in Vancouver, which runs one of the world's largest cyclotrons for nuclear and particle physics and produces medical isotopes; the Canadian Light Source in Saskatoon, a synchrotron used by physicists, chemists and biologists; and SNOLAB near Sudbury. Whatever the scale, the same principles apply: understand your instrument, quantify your uncertainties, and look for what could fool you.