In Photons and the Photoelectric Effect, light of six known frequencies falls on a sodium surface. For each frequency a voltmeter gives the stopping potential. Einstein's equation says the six points lie on a straight line whose slope is \(h/e\). But the six readings do not lie exactly on any one line. A ruler can be laid through them in many ways. Which line is the right one?
Axes and scales
A graph of readings begins with two choices. The quantity that the experimenter sets, here the frequency \(\nu\), goes along the x axis. The quantity that responds, here the stopping potential \(V_0\), goes up the y axis. Each axis carries the name of its quantity and its unit, such as \(V_0\) (V).
The scales are then chosen so that the readings fill most of the graph. One division stands for a simple step, such as 1, 2 or 5 units. A point can then be placed and read without awkward arithmetic. A graph squeezed into one corner wastes the paper and hides the slope.
The frequencies run only from \(7\) to \(12 \times 10^{14}\,\text{Hz}\). Starting the axis at zero spends more than half the width on frequencies never measured. So the axis starts at \(6 \times 10^{14}\,\text{Hz}\), a false origin, shown by a small break in the axis. The price is that the intercept on the \(V_0\) axis, at \(\nu = 0\), is now off the graph. It must be calculated instead of read.
A student plots load against extension for a wire, with loads from \(10\) to \(40\,\text{N}\) and extensions from \(0.80\) to \(3.20\,\text{mm}\). Should she use a false origin?
A line among the points
Each reading carries a random error, so the points scatter about the true line. The best line is the one that the scatter fits most evenly. Lay a transparent ruler on the graph and turn it until the points lie as much above it as below, along its whole length. The line need not pass through any single reading.
Two facts from the figure above guide the eye. The best line passes through the mean point, G, whose coordinates are the mean of the x values and the mean of the y values. And when the line is turned too far either way, the gaps at the two ends grow and all lie on one side. At the best slope the gaps are small and alternate in sign.
The best straight line through a set of readings passes through their mean point. It leaves the readings scattered evenly on both sides along its length. Its slope is found from two points on the line that are far apart.
Once the line is drawn, the readings have done their work. Take two points on the line itself, near its two ends, and read their coordinates. Two readings would carry their own random errors; two points on the line carry the average of all six. Points far apart also make the differences \(y_2 - y_1\) and \(x_2 - x_1\) large compared with the error of reading the graph.
The figure above is a picture of the method of least squares. The best line is the one that makes the sum of the squares of the gaps smallest, 0.0008 V² here. Adrien-Marie Legendre published the method in 1805, and Gauss used it to find the orbits of planets. A calculator's "linear regression" does exactly this. The line drawn by eye through the mean point comes very close to it.
Why is the slope found from two points on the drawn line, and not from the first and the last readings?
What the slope and the intercept mean
Compare the law with the equation of a line. Einstein's equation, divided by \(e\), reads \(V_0 = (h/e)\,\nu - \phi/e\). This has the form \(y = mx + c\). So the slope \(m\) is \(h/e\), in volts per hertz, and the intercept \(c\) is \(-\phi/e\), in volts. Every slope and every intercept of a graph of readings is a physical quantity with a unit.
This is the worked example of Photons and the Photoelectric Effect, lesson 4, done with the tool. The best line through the six readings passes through \(0.57\,\text{V}\) at \(8.0 \times 10^{14}\,\text{Hz}\) and \(2.22\,\text{V}\) at \(12.0 \times 10^{14}\,\text{Hz}\). Find \(h\), the threshold frequency and the work function. Take \(e = 1.6 \times 10^{-19}\,\text{C}\).
Two points on the line, far apart: \((8.0 \times 10^{14}, 0.57)\) and \((12.0 \times 10^{14}, 2.22)\).
\(m = (2.22 - 0.57)/(4.0 \times 10^{14}) = 4.125 \times 10^{-15}\,\text{V s}\), so \(h = e\,m = 6.60 \times 10^{-34}\,\text{J s}\).
The false origin hides the intercept, so calculate it: \(c = 0.57 - 4.125 \times 10^{-15} \times 8.0 \times 10^{14} = -2.73\,\text{V}\). The line meets \(V_0 = 0\) at \(\nu_0 = 2.73/(4.125 \times 10^{-15}) = 6.62 \times 10^{14}\,\text{Hz}\).
\(\phi = -e\,c = 2.73\,\text{eV}\), close to sodium's \(2.75\,\text{eV}\), as the physics chapter finds.
Another student's best line of \(V_0\) against \(\nu\), on a false origin, passes through \(0.15\,\text{V}\) at \(7.0 \times 10^{14}\,\text{Hz}\) and \(2.20\,\text{V}\) at \(12.0 \times 10^{14}\,\text{Hz}\). Calculate the intercept at \(\nu = 0\) and give the work function, in eV.
Making a law straight
Many laws are not straight lines, but a change of variable makes them straight. For a pendulum \(T = 2\pi\sqrt{L/g}\), so \(T\) against \(L\) is a curve. Square both sides: \(T^2 = (4\pi^2/g)\,L\). Now \(T^2\) against \(L\) is a straight line through the origin, with slope \(4\pi^2/g\). A straight line is easy to judge by eye, and a bend in it warns at once that the law has failed.
In a school laboratory, not in a physics chapter, a student plots \(T^2\) against \(L\) for a pendulum. Her best line has a slope of \(4.02\,\text{s}^2/\text{m}\). What value of \(g\) does it give, in m/s²?
In Searle's experiment the best line of load against extension passes through \(40\,\text{N}\) at \(3.20\,\text{mm}\). The wire is \(2.00\,\text{m}\) long with an area of \(1.257 \times 10^{-7}\,\text{m}^2\). Find Young's modulus, in units of \(10^{11}\,\text{Pa}\).
A line of readings holds only where readings were taken. Below the threshold frequency no electrons leave the metal, so the dashed part of the photoelectric line is arithmetic, not a measurement. In the same way the load-extension line of a wire bends once the elastic limit is passed. Extend a line only to calculate an intercept, and never trust it as a prediction far beyond the readings.
The best straight line turns a set of readings into a constant of nature. In Photons and the Photoelectric Effect its slope gives Planck's constant and its intercept the work function. In Searle's experiment the slope of load against extension gives Young's modulus, and the line averages the random errors of the readings. The same method gives \(g\) from \(T^2\) against \(L\). It also gives a resistance from the slope of V against I.
Back to the voltmeter
The six readings of sodium do not lie on one line, and no line should be forced through two of them. The best line passes through their mean point with the scatter balanced on both sides. Its slope, \(4.125 \times 10^{-15}\,\text{V s}\), gives \(h = 6.60 \times 10^{-34}\,\text{J s}\). Its calculated intercept, \(-2.73\,\text{V}\), gives a work function of \(2.73\,\text{eV}\). A voltmeter and a set of filters have measured a constant of nature.
- Joining the points dot to dot, or forcing the line through the first and the last reading.Draw one straight line with the scatter balanced on both sides, through the mean point.
- Reading the slope from two readings.Take two points on the drawn line, far apart.
- Reading the intercept from a graph with a false origin.The axis there is not at zero. Calculate \(c = y_1 - m x_1\).
- Giving a slope without its unit.The slope has the unit of y per unit of x: V per Hz, N/mm, s²/m.