In a college laboratory, a student shines light through five coloured filters, one after another, onto a photocell. For each colour she finds the retarding voltage that just stops the current. She plots the five voltages against the frequencies of the light, and the points lie on a straight line. From the tilt of that line and the places where it meets the axes, she can find a constant of nature.
Equal steps, equal rises
A train of the Delhi Metro passes a signal \(100\,\text{m}\) from a station and runs on at a steady \(20\,\text{m/s}\). Its distance from the station is \(x = 100 + 20t\), with \(t\) in seconds. Every second adds the same \(20\,\text{m}\). A graph of \(x\) against \(t\) is therefore a straight line. Equal steps in \(t\) give equal rises in \(x\).
Any straight line can be written \(y = mx + c\). The number \(m\) is the slope. It is the rise in \(y\) divided by the run in \(x\), and it is the same between any two points of the line. The number \(c\) is the intercept. It is the value of \(y\) where the line crosses the \(y\) axis, at \(x = 0\). For the train, \(m = 20\,\text{m/s}\) and \(c = 100\,\text{m}\).
The slope of a straight line is the rise divided by the run between any two of its points, \(m = \Delta y/\Delta x\). The intercept \(c\) is the value of \(y\) where the line crosses the \(y\) axis. A straight line is the graph of \(y = mx + c\), and its slope is the same everywhere on it.
A positive slope means that \(y\) grows as \(x\) grows, so the line climbs to the right. A negative slope means that \(y\) falls, so the line drops to the right. A zero slope gives a flat line, \(y = c\). A steeper line has a slope of larger size. Two lines with the same slope never meet, because they are parallel.
A cyclist's position-time graph is a straight line through \((2\,\text{s}, 10\,\text{m})\) and \((6\,\text{s}, 30\,\text{m})\). What is its slope, in m/s?
The units name the physics
A slope is a rise divided by a run. So its unit is the unit of \(y\) divided by the unit of \(x\). An intercept is a value of \(y\), so it has the unit of \(y\). Ask for the unit of the slope, and you usually learn what the slope measures. On a position-time graph the slope is in m/s, a velocity. On a velocity-time graph it is in m/s², an acceleration.
Where the line meets the horizontal axis
The line meets the \(x\) axis where \(y = 0\). Put \(y = 0\) into \(y = mx + c\) to get \(x = -c/m\). This point often has its own physical name. On the stopping-potential line it is the threshold frequency. On the line of a cell it is the short-circuit current. On the volume-temperature line of a gas it is \(-273\,^\circ\text{C}\), the absolute zero.
A straight line is not always a proportion
The quantity \(y\) is proportional to \(x\) only when \(c = 0\). Then the line passes through the origin, and doubling \(x\) doubles \(y\). If \(c\) is not zero, doubling \(x\) does not double \(y\). A Fahrenheit reading is \(F = 1.8C + 32\). It is \(68\) at \(20\,^\circ\text{C}\), but \(104\), not \(136\), at \(40\,^\circ\text{C}\).
The Fahrenheit reading is \(F = 1.8C + 32\). A thermometer reads \(68\,^\circ\text{F}\) at \(20\,^\circ\text{C}\). What does it read at \(40\,^\circ\text{C}\)?
At what temperature do a Celsius thermometer and a Fahrenheit thermometer show the same number? Use \(F = 1.8C + 32\) and give the reading.
A straight line fitted to measurements is known only over the range that was measured. The stopping-potential line exists only above the threshold frequency; its dashed extension down to \(\nu = 0\) is arithmetic, not a measurement. A line for the resistance of a copper wire, measured from \(0\) to \(100\,^\circ\text{C}\), cannot be trusted near \(-250\,^\circ\text{C}\). Use the extension to read an intercept, never as a result of the experiment.
A line from one point and its slope
Often you know one point of a line and its slope, but not its intercept. Any other point \((x, y)\) of the line must make the same slope with the known point \((x_1, y_1)\). This one fact gives the equation of the line. The derivation below takes three steps.
\[ \dfrac{y - y_1}{x - x_1} = m \]
Why. the slope between the known point \((x_1, y_1)\) and any other point \((x, y)\) of the line is the slope \(m\) of the line.
\[ y - y_1 = m(x - x_1) \]
Why. multiply both sides by \(x - x_1\). This form also holds at the known point itself, where both sides are zero.
\[ y = mx + (y_1 - mx_1) \]
Read it. open the bracket. This is \(y = mx + c\) with the intercept \(c = y_1 - mx_1\).
A line has slope \(3\) and passes through the point \((2, 5)\). What is its equation?
Back to physics: the photoelectric line
Einstein's equation for the photoelectric effect is \(eV_0 = h\nu - \phi\). Divide by \(e\) to get \(V_0 = (h/e)\,\nu - \phi/e\). This has the form \(y = mx + c\) with \(y = V_0\) and \(x = \nu\). So the slope is \(h/e\), and the intercept on the \(V_0\) axis is \(-\phi/e\). The line meets the frequency axis at the threshold frequency \(\nu_0\).
The best line of \(V_0\) against \(\nu\) 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 Planck's constant, the threshold frequency and the work function. Take \(e = 1.6 \times 10^{-19}\,\text{C}\).
Two points of the line: \((x_1, y_1) = (8.0 \times 10^{14}, 0.57)\) and \((x_2, y_2) = (12.0 \times 10^{14}, 2.22)\). The slope is \(h/e\), and the line meets the frequency axis at \(\nu_0\).
\(m = (y_2 - y_1)/(x_2 - x_1)\), then \(h = e\,m\). For the threshold, put \(V_0 = 0\) into the point-slope form \(V_0 - 0.57 = m(\nu - 8.0 \times 10^{14})\).
\(m = 1.65/(4.0 \times 10^{14}) = 4.125 \times 10^{-15}\,\text{V s}\), so \(h = 1.6 \times 10^{-19} \times 4.125 \times 10^{-15} = 6.60 \times 10^{-34}\,\text{J s}\). At \(V_0 = 0\), \(\nu_0 = 8.0 \times 10^{14} - 0.57/(4.125 \times 10^{-15}) = 6.62 \times 10^{14}\,\text{Hz}\).
\(\phi = h\nu_0 = 6.60 \times 10^{-34} \times 6.62 \times 10^{14} = 4.37 \times 10^{-19}\,\text{J} = 2.73\,\text{eV}\). The intercept gives the same: \(c = 0.57 - 4.125 \times 10^{-15} \times 8.0 \times 10^{14} = -2.73\,\text{V} = -\phi/e\).
The student of the laboratory now has her constant. The tilt of her line is \(h/e\), so a voltmeter and a set of filters have measured \(h\). The place where the line starts, on the frequency axis, is the threshold of the metal. The extension of the line back to zero frequency gives the work function.
A cell gives a terminal voltage of \(5.5\,\text{V}\) while it supplies \(1\,\text{A}\) and \(5\,\text{V}\) while it supplies \(2\,\text{A}\). Its graph of \(V\) against \(I\) is the straight line \(V = \mathcal{E} - Ir\). Find its emf, the intercept of the line, in volts.
What is the unit of the slope of a graph of stopping potential \(V_0\), in volts, against frequency \(\nu\), in hertz?
The straight line is the most used graph in the book. The chapter Graphical Understanding of Motion reads the acceleration as the slope of a v-t line. Ohm's law makes the V-I graph of a metal a line through the origin. The cell's terminal voltage falls along a line of slope \(-r\). The photoelectric chapter reads Planck's constant from a slope. Moseley's law makes \(\sqrt{\nu}\) of an X-ray line a straight line against the atomic number. Thermometer scales, the absolute zero, and the mirror and lens graphs of \(1/v\) against \(1/u\) are straight lines too.
- Taking a line with an intercept as a proportion.Only a line through the origin is a proportion. With \(c \ne 0\), doubling \(x\) does not double \(y\).
- Finding the slope from one point, as \(y/x\).The slope is a rise over a run between two points, \((y_2 - y_1)/(x_2 - x_1)\). The ratio \(y/x\) equals it only when \(c = 0\).
- Treating the extended part of a fitted line as data.The extension is used to read an intercept. It is not a measurement; below the threshold there is no stopping potential at all.