In the late 19th century, J. J. Thompson began working with a device known as a cathode ray tube (CRT) to investigate the nature of the mysterious “cathode rays” that the device produced.

When scientists passed electricity through a glass tube filled with a low-density gas, they saw a beam of glowing light – the so-called cathode rays. Thompson discovered that he could deflect the path of the beam by applying a magnetic field, and in doing so showed that the beam was a stream of negatively charged particles – what we today know are electrons. (The explanation for why the gas glowed when the energetic electrons passed through it would not become clear until quantum mechanics explained the ideas of gas ionization and transitions of electrons between energy levels… but that will have to wait for another experiment.)

Thompson's experiment is considered the “discovery” of the electron, but it could determine neither the magnitude of the charge nor the mass of the particle… only the ratio of the two: $e/m$. It was not until Millikan's oil drop experiment – which you replicated earlier this quarter – established the value of the fundamental charge e that Thompson's ratio could be solved for the mass of the electron on its own.

This section is to be completed individually BEFORE your first meeting with the TA and your lab group.

This experiment will be a little bit different than the previous ones we've done this year. This time, we will show a real-life apparatus and ask you to consider how it could be used to determine the charge to mass ratio, $e/m$. After understanding the apparatus, we will provide some data collected on the device which you (and your group) will analyze and discuss.

Click on the link below to start your individual lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.)

Remember to fill this out as you go along. Do not wait until you have completed the experiment. The notebook is meant to be a record of everything you've done in lab – good and bad – and is only useful if it is complete and honest.

The main piece of the apparatus is the cathode ray tube (CRT). In our setup, the CRT is under vacuum (with no gas inside) and contains the following basic components inside (which are shown in Fig. 1):

• the electron gun, used to create and accelerate the electrons to known energy,
• an electrostatic deflection system, (not used in today's experiment), and
• a fluorescent screen, which glows green when struck by a high energy electron.

Figure 1: Schematic of the cathode ray tube

The tube operates as follows:

• A thin tungsten wire and a surrounding metal sheath called a cathode are heated by an electric current passing through the wire. This hot cathode emits electrons.
• Around the cathode is a metal cylinder with a small hole in its end. This cylinder is called the control grid, since its primary purpose is to control the electron current emitted from the cathode, flowing into the acceleration region of the electron gun.
• Following the control grid are the anodes which accelerate the electrons, and focus them onto the screen.
• Following the accelerating anodes, there are two sets of deflection plates, one deflecting the beam vertically, and the other deflecting the beam horizontally. (These are not used for this experiment.)
• Finally, there is a screen which fluoresces when struck by the energetic electron beam.

The cathode ray tube is the basic component of analog oscilloscopes and older-style TV picture tubes.

For the case of an electron of charge $e$ moving through the accelerating voltage $V_a$, we may use conservation of energy to relate its final kinetic energy to the initial potential energy as

where $m$ is the mass of the electron and $v$ is its final velocity. After the electron is accelerated to this final velocity, it will continue in a straight path through the rest of cathode ray tube and strike the center of the fluorescent screen making a green dot (unless we do something to deflect it).

To deflect the beam of electrons, we can apply a magnetic field. A particle of charge $q$ moving at velocity $\textbf{v}$ in a magnetic field $\textbf{B}$ will feel a Lorentz force

or, in scalar form, a force of magnitude

where $\theta$ is the angle between the velocity and magnetic field vectors. For our experiment, we will place our magnetic field perpendicular to the motion of the electrons (i.e. $\sin\theta = 1$) and the electron charge is of course $q=e$, which means Eq. (3) simplifies to

Some pictures of the apparatus are shown in Fig. 2. The CRT is contained within a plastic box (to protect it from breakage) and a coil of wire is wound to make a loop on either side of the tube. When we pass current through those wires, the two loops will produce a magnetic field perpendicular tube.

Figure 2: Photos of the apparatus: (top) view from above, (middle) side view, (bottom) view from front (including meters).

Figure 3 shows a side view of the CRT and the magnetic field-producing coils. Figure 3 also shows the measured magnetic field profile along the tube length. The maximum magnetic field is normalized to 1.0, and so we can see that the field is not quite uniform over the length of the tube. (It falls off a little bit in strength at the front and back of the CRT tube.)

Figure 3: Coil geometry and magnetic field map.

The relation between the current in the coils and the magnetic field along the electron beam trajectory is

So how do we determine the charge to mass ratio using this setup?

As already mentioned, our electrons are produced and accelerated up to a final velocity $v$ – given in Eq. (1) –  before entering a drift region where they feel a Lorentz force – given by Eq. (4). This force will cause the electron path to curve, and when the electron strikes the fluorescent screen it will be displaced from the original center spot by some distance $D$ as shown in Fig. 4. (Why do we show an upward deflection for the electron if the magnetic field is directed out of the page?)

Figure 4a: Electron trajectory geometry

To determine the amount of the deflection, we have to do a little bit of geometry. The electron beam will follow a circular trajectory within the field with a centripetal force

where $R$ is the radius of the circle. Using our force from Eq. (4) and our velocity $v^2$ from Eq. (1), we have

Figure 4b: Electron trajectory geometry, highlighting the relevant similar triangles

Since $R$ is not measurable in this experiment, we wish to express it in terms of other variables which are measurable. Referring to Fig. 4b, by similar triangles (dashed green triangle and dotted white triangle) we have

However, $\overline{AB} = \overline{OB}/2$. Therefore,

For small angles $\theta$ (such as those in our cathode ray tube),

Therefore,

Substituting this expression for $R$ into Eq. (7) gives

Since one can measure all of these quantities, it is now possible to arrive at a value of $e/m$.

Equation (8) is derived using the following simplifying assumptions:

• The magnetic field is assumed to be perfectly constant over the well-defined path length $L$.
• The electrons are assumed to be moving at constant speed along $L$.

In our experiment, however, $B$ is not perfectly constant over the electron beam trajectory and $L$ is not well-defined. Also, the electrons are not moving at constant speed for the first 4 cm of travel.

Looking back at Fig. 3, we see the following:

• the electrons do not begin accelerating at position $x = 0$,
• the electrons are accelerating (but have not yet reached full velocity) in the region between about $x$ = 4 cm and $x$ = 8 cm, and
• the electrons experience a decreasing magnetic field over the final region from about $x$ = 20 cm to $x$ = 23 cm.

The range of possible values for $L$ is therefore between 15 cm and 19 cm. This will be an important point that we will return to during the analysis.

There are two separate circuits required to power the apparatus (as shown in Fig. 5). The left side shows the wiring for the cathode ray tube which produces the beam of electrons and accelerates them toward the tube’s screen. The right side shows the circuit which produces the magnetic field which will deflect the electron beam.

Figure 5: Wiring diagram showing accelerating voltage connections (left) and magnetic field connections (right)

To further clarify how the apparatus is setup and operates, we also provide a video of the wiring and setup in Fig. 6.

You are not physically in the lab to take data, but we've tried to recreate the experience for you here.

Below are several photos of the deflection spot under different conditions.

• Look at each and measure the deflection from the center point, $D$. (How can you determine the center point?)
• Estimate the uncertainty in the deflection, $\delta D$.

Click on images to enlarge.

NOTEBOOK: Look through the images. Determine the position of the center point (from the appropriate image) and then determine the displacement $D$ and uncertainty $\delta D$ for each other image. How do you determine this uncertainty?

NOTEBOOK: Are there other parameters (measured or given) where there is significant uncertainty?  

In Part 2, you and your group will split up the remaining data processing and analyze it together. This is just a little bit of practice before the meeting so that you can ask questions about the process if you have difficulty.

At the group discussion, you will have an opportunity to ask questions about the setup and about the preliminary data images you tried to measure  with. Your TA will lead a discussion about uncertainties and help you plan your data collection and analysis strategy for part 2 of the project.  

NOTEBOOK: Take notes during your meeting. What did your group talk about? What results (from you or your groupmates) are important to keep in mind? What information did the TA provide to guide you? What did your group decide to do next in Part 2?

This section is to be completed AFTER your first meeting with the TA and your lab group.

For the second part of this project, your group will have access to a catalog of videos of the apparatus in action. From these videos, you can extract displacement measurements (e.g. by pausing the video and making measurements off the still screen) for use in your analysis. Your goal is both to determine an estimate for the charge to mass ratio e/m and to test the applicability of the model given by Eq. (8) over a range of parameters (e.g. for different values of $V_a$, $B$, etc.)

The available data is listed in Table 1 (lower resolution) and Table 2 (higher resolution). Note that some videos have been taken while holding the accelerating voltage constant (and changing magnetic field strength), while other videos have been taken while holding magnetic field strength constant (and changing accelerating voltage). You do not need to use all the videos provided. You and your group should decide what subset to use (and why), and decide how to divide up the work amongst yourselves. It may be useful, for example, to have some data that everyone looks at (as a control to ensure consistency across the group) and to have a larger selection of videos that are each only processed by one student (in order to collect more data overall).

Once you have your data, you can try plotting it to see what it looks like. You can use a program like Microsoft Excel or Google Sheets, or you can use a free online grapher like Desmos. (Or if you are comfortable, you can also use this customized Google Colab notebook:

• How should you plot your data? (Which variable is on the $x$-axis? Which is on the $y$-axis? Which variables do you keep constant?)
• When you plot your data, where does $e/m$ appear? (Is it the slope of a line? The $y$-intercept? Something else?)
• Does your data obey the trend you expect? (For example, if you expect the data to be linear, is it? Are there outliers?)

We aren't looking for you to do anything sophisticated yet… just look at your data and decide if it looks reasonable or not. Make a note of any questions you have to bring up at your meeting with your TA.

At the group discussion, your TA will pull together everyone's data, fit it to expected forms, and hold a discussion about how to interpret the results.

During the meeting, your TA may ask you to look at the following discussion topics, or you may refer to this information after the meeting for a reminder of certain definitions or explanations.

After your second meeting, you will again need to write up your summary and your conclusions. Include any data tables, plots, etc. from the experiment or discussions as necessary in order to show how your data support your conclusions.

This part doesn't need to be long; one or two pages should be sufficient. What is important, however, is that your writing should be complete and meaningful. Address both the qualitative and quantitative aspects of the experiment, and make sure you cover all the “take-away” topics in enough depth. Don't include throw-away statements like “Looks good” or “Agrees pretty well.” Instead, try to be precise.

Remember… your goal is not to discover some “correct” answer. In fact, approaching any experiment with that mind set is the exact wrong thing to do. You must always strive to reach conclusions which are supported by your data, regardless of what you think the “right” answer should be. Never, under any circumstances should you state a conclusion which is contradicted by the data. Stating that the results of your experiment are inconclusive, or do not agree with theoretical predictions is completely acceptable if that is what your data indicate. Trying to shoehorn your data into agree with some preconceived expectation when you cannot support that claim is fraudulent.