Fotoni lab visit: modern physics experiments

Before the visit: the physics of each station

Fotoni is the open physics teaching laboratory of the University of Helsinki. Its material bank describes the stations below. Your teacher chooses with Fotoni which ones your group does, and the equipment on the day can differ a little from the instructions: always check with the instructor.

Photoelectric effect: Planck's constant

LEDs of different colours shine on the metal plate of a vacuum tube (a photocell). A counter-voltage pushes the freed electrons back. You raise it until the current is zero: this is the stopping voltage $U_\text{s}$. From the unit Light as particles: the photoelectric effect you know that ${eU_\text{s} = hf - W}$. You measure $U_\text{s}$ for every LED and find Planck's constant $h$ from the slope of a graph.

The charge-to-mass ratio of the electron

The magnetic force is the centripetal force, and the voltage gives the kinetic energy:

$$\begin{aligned} evB &= \frac{mv^2}{r} \\ eU &= \tfrac{1}{2}mv^2 \end{aligned}$$

Together these give ${2U = (e/m) \cdot B^2r^2}$. The field of the coil pair is ${B = (4/5)^{3/2}\,\mu_0 N I / R}$, where $N$ is the number of turns of one coil, $R$ its radius and $I$ the current.

Double slit, also with single photons

Light passes two narrow slits, and a detector moves across the pattern. Bright light is measured with a photodiode. Then a very dim green lamp is used, so that there is practically never more than one photon in the apparatus. A photomultiplier turns each photon into a pulse, and a counter counts the pulses in $10\ \text{s}$. This links to the unit Waves, probability and quantum tunnelling.

Hot filament and spectra

The light of a lamp filament is spread out by a prism, and a sensor measures the intensity at each wavelength at four voltages ($7$, $8$, $9$ and $10\ \text{V}$). The software calculates the filament's temperature. Compare with Wien's law ${\lambda_\text{max} T = b}$ (unit When classical physics ran out of answers). At the spectra station you look at lamps and gas tubes with a hand spectroscope and a spectrometer (unit From Rutherford's nucleus to Bohr's atom).

Millikan's oil drops (qualitative)

Tiny oil drops float between two plates with a voltage of about $500\ \text{V}$. An alpha source charges them. You watch the drops through a microscope camera and look for sudden changes in their motion.

Planning a measurement

Every station has a research question: what do you want to find out? Then name the variables:

• the independent variable is the one you change (the LED, the coil current);
• the dependent variable is the one you measure (the stopping voltage, the radius);
• the controlled variables are kept the same (the accelerating voltage, the intensity).

No measurement is exact. The uncertainty $\Delta x$ of a quantity $x$ tells how far the true value may be from your reading. For a scale or a digital display, the reading uncertainty is about half of the smallest step you can read, unless the reading jumps around more. If you repeat a measurement, use the mean. Half of the range, (largest minus smallest) divided by two, is a simple estimate of how much single readings spread; use it as $\Delta x$ of one reading. It is not the uncertainty of the mean, which is smaller. Write the result as ${x \pm \Delta x}$ with its unit. The relative uncertainty $\Delta x / x$ is often given in percent.

Safety at the stations

• Several stations use high voltages: the electron tube up to $300\ \text{V}$, the oil-drop plates about $500\ \text{V}$, and the gas tubes. Change the connections only when the instructor allows it.
• The electron tube is thin glass: a magnet may come near it, but never touch it.
• Lamps and gas tubes get hot. Do not touch them, and do not look straight into a bright lamp or LED from close by.
• The alpha source of the oil-drop station stays inside the apparatus. Only the instructor handles it.
• Do not run the filament lamp at $10\ \text{V}$ for long: it shortens the lamp's life.

During the visit: lab worksheet

Print this section, or copy the tables into your notebook. Fill in the empty cells at the station. Write every number with its unit and its uncertainty.

Group

Names	Date	Stations we did

Station: photoelectric effect

Research question: how does the stopping voltage depend on the frequency of the light, and what is Planck's constant?

Instrument at the station	What it measures	Smallest step	$\Delta x$
	voltage $U$
	current $I$

Observations with the red LED, before you take the measurements:

What we did	What happened to the current
$U$ slightly negative, more light
$U$ at the stopping voltage, more light

LED colour	$\lambda\ (\text{nm})$	$f\ (10^{14}\ \text{Hz})$	$U_\text{s}\ (\text{V})$
red

Where does the LED's wavelength come from (a label, the instructor, a spectrometer)? Write it down.

Station: charge-to-mass ratio of the electron

Research question: what is the ratio $e/m$ of the electron?

Coil data	Value	Where it comes from
turns of one coil $N$
coil radius $R\ (\text{m})$

Observations before the measurements:

What we did	What happened to the beam
a bar magnet near the tube (not touching)
$U$ kept the same, coil current up
coil current kept the same, $U$ up

Measurements:

$I\ (\text{A})$	$U\ (\text{V})$	$r\ (\text{m})$

Calculated from each row of the table above:

$B\ (\text{T})$	$B^2r^2\ (\text{T}^2\text{m}^2)$	$2U\ (\text{V})$

How did you measure the radius, and what is $\Delta r$?

Station: double slit and single photons

Research question: does a pattern of bright and dark bands appear when photons arrive one at a time?

Data of the setup	Value	Where it comes from
slit separation $d\ (\text{mm})$
slit to detector $L\ (\text{cm})$
wavelength $\lambda\ (\text{nm})$

Repeated counts at one position (each in $10\ \text{s}$):

Count 1	Count 2	Count 3	Count 4	Count 5

Counts at different positions:

Position $x\ (\text{mm})$	Counts in $10\ \text{s}$	Position $x\ (\text{mm})$	Counts in $10\ \text{s}$

Station: hot filament

Research question: how does the spectrum of a hot filament change with its temperature?

Voltage $U\ (\text{V})$	Temperature $T\ (\text{K})$	Peak $\lambda_\text{max}\ (\text{nm})$	Colour we saw
$7$
$8$
$9$
$10$

Station: spectra

Research question: which light sources give a continuous spectrum, and which give lines?

Light source	Continuous or lines?	Brightest colours or lines

Station: oil drops

Research question: how do charged oil drops move between the plates, and what happens when a drop's charge changes?

What we saw	Our explanation

After each station

One sentence: what surprised us, and what limited the precision?

Station	Our sentence

A straight-line fit

When a law has the form $y = kx + b$, plot the measured points $(x, y)$ and fit the straight line that passes closest to all of them. A spreadsheet or a graphing calculator does this (a trendline or linear fit). For the photocell, ${U_\text{s} = (h/e)\,f - W/e}$, so

$$\begin{aligned} k &= \frac{h}{e}, & h &= e\,k, \\ b &= -\frac{W}{e}, & W &= -e\,b. \end{aligned}$$

For the electron beam, ${2U = (e/m) \cdot B^2r^2}$ is a line through the origin, and its slope is $e/m$.

The uncertainty of the slope can be estimated with the steepest and least steep lines that still pass through the uncertainty ranges of the points, usually the first and the last point. Then

$$\Delta k \approx \frac{k_\text{max} - k_\text{min}}{2}$$

Finally, compare your result with the accepted value by the percentage difference:

$$\text{difference} = \dfrac{x - x_\text{acc}}{x_\text{acc}} \cdot 100\ \%$$

where $x$ is your result and $x_\text{acc}$ the accepted value.

If the accepted value lies inside ${x \pm \Delta x}$, your result agrees with it within the uncertainty.

Worked example

Writing a lab report in English

A lab report tells a reader what you did, what you found and how sure you are. Scientists write it in a fixed order:

1. Title and aim: the research question in one sentence.
2. Background: the physics and the equation you test, in the present tense ("The stopping voltage depends on …").
3. Method: the equipment and what you did, in the past tense ("We measured …", or the passive "The voltage was increased …").
4. Results: tables with units in the headings, a graph with labelled axes and the fitted line.
5. Analysis: the fit, the result with its uncertainty, ${x \pm \Delta x}$.
6. Discussion: the comparison with the accepted value, the sources of uncertainty and how to improve the experiment.
7. Conclusion: the answer to the research question in one or two sentences.

Write numbers with a decimal point and their unit ($0.93\ \text{V}$). Do not claim more than the data shows: "the result agrees with the accepted value within its uncertainty" is stronger science than "the result proves the theory".

Common misconception

"An uncertainty means that we made a mistake." No: every measurement has an uncertainty, even a perfect one. A mistake (a wrong connection, a misread scale) is something you can find and correct. The uncertainty estimates how far the true value may be from the result. Part of it shows as scatter when you repeat the measurement. Systematic errors, such as the band of wavelengths of an LED, do not show as scatter, and repeating does not reduce them.

A second misconception is that more decimals mean a better result. A calculator may show ${6.351596 \cdot 10^{-34}}$, but with ${\Delta h = 0.3 \cdot 10^{-34}\ \text{J} \cdot \text{s}}$ only the first decimal means something. A third is that a result that differs from the accepted value is wrong. It is fine if the accepted value lies inside ${x \pm \Delta x}$.

Summary

• At Fotoni you measure modern physics yourself: Planck's constant from stopping voltages, $e/m$ from a circular electron beam, single photons in a double slit, the spectrum of a hot filament, line spectra and charged oil drops.
• Plan every measurement: research question, independent, dependent and controlled variables, and the uncertainty $\Delta x$ of each reading.
• The slope of a fitted line gives the constant: ${h = e\,k}$ from ${U_\text{s} = (h/e)\,f - W/e}$, and $e/m$ from ${2U = (e/m) \cdot B^2r^2}$.
• The steepest and least steep lines give ${\Delta k \approx (k_\text{max} - k_\text{min})/2}$. Report ${x \pm \Delta x}$ and the percentage difference from the accepted value.
• A lab report has a fixed structure: aim, background, method, results, analysis, discussion and conclusion.