Magnetic Fields I: $e/m$ of Electrons

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 – performed between 1900 and 1911 here at the University of Chicago – established the value of the fundamental charge $e$ that Thompson's ratio could be solved for the mass of the electron on its own.

Theory


The force on a charge moving in a magnetic field is

$\mathbf{F} = q\mathbf{v}\times\mathbf{B}$, $(1)$

where $q$ is the charge, $\mathbf{v}$ is the velocity of the moving charge, and $\mathbf{B}$ is the magnetic field. The direction of the force is given by the right-hand rule and is perpendicular to both the velocity and magnetic field.

The magnitude of the force is given by the scalar form of Eq. (1),

$F = qvB\sin\phi$, $(2)$

where $\phi$ is the angle between the direction of the magnetic field and the direction of motion of the moving charge.

Suppose a beam of electrons is directed into a magnetic field at right angles to the field as shown in Fig. 1.

Figure 1a: Electron trajectory geometry

In this special case, Eq. (2) becomes

$F = evB$, $(3)$

where $e$ is the charge of the electron. The electron beam will follow a circular trajectory within the field with a centripetal force

$F=\dfrac{mv^2}{R}$, $(4)$

where $m$ is the mass of the electron and $R$ is the radius of the circle.

For a non-relativistic electron accelerated through a potential $V$, the kinetic energy is

$K = \frac{1}{2}mv^2 = eV$. $(5)$

Setting Eqs. (3) and (4) equal, solving for $v$, and substituting into Eq. (5), we have

$\dfrac{e}{m} = \dfrac{2V}{R^2B^2}$. $(6)$

Figure 1b: 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. 1b, by similar triangles (dashed green triangle and dotted white triangle) we have

\begin{equation*} \dfrac{\overline{AB}}{R} = \dfrac{D}{\overline{OB}}. \end{equation*} However, $\overline{AB} = \overline{OB}/2$. Therefore, \begin{equation*} R = \dfrac{\left(\overline{OB}\right)^2}{2D}. \end{equation*} For small angles $\theta$ (such as those in our cathode ray tube), $\overline{OB} \approx L$. Therefore, \begin{equation*} R = \dfrac{L^2}{2D} \end{equation*} Substituting this expression for $R$ into Eq. (6) gives

$\dfrac{e}{m} = \dfrac{8V_aD^2}{L^4B^2}$. $(7)$

Since one can measure all of these quantities but $L$, it is now possible to arrive at a value of $e/m$. To make this more amenable to plotting, we may rearrange this as

$D^2 = \dfrac{e}{m} \dfrac{L^4B^2}{8V_a}$. $(8)$

In this way, the ratio $e/m$ is proportional to the slope of a line plotting either $D^2$ vs. $B^2$ or $D^2$ vs. $1/V_A$.

Experimental setup


Lab notebook template

One member of the group should click on the link below to start your group lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.) Make sure to share the document with everyone in the group (click the “Share” button in the top right corner of the screen) so each member has access to the notebook after you leave lab.

Constructing the circuits

Construct the two separate parts of the apparatus as shown in Fig. 2. The portion of each schematic  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.

Physics 132

Figure 2a: $e/m$ wiring diagram using Heathkit power supply model IP-17
Physics 122

Figure 2b: $e/m$ wiring diagram using 3B Scientific power supply

When wiring your setup, you should start by connecting the yellow wires, and then connected to the $+$ terminal of your supply.
Click to enlarge either diagram

Wiring photos & connections


Photo of the red wire, brown wire, and yellow paired wires from the electron tube. Note that the red and brown wires do not stack, they will have to be plugged in last.

Photo of the twin leads from the electron tube. Each controls one of the deflection plates. While we won't be using the plates, these still need to be grounded to prevent unwanted behavior.
Physics 132
Physics 122
Photo of Coil connections around the cathode ray tubes. Note that the 132 course uses connectors on the back right of the setup, whereas the 122 course uses connectors beneath the front of the tube.

Magnetic field

Equation (7) giving $e/m$ for the electron from the measured quantities is derived using the following simplifying assumptions:

  • The magnetic field $B$ 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 $4\text{ cm}$ of travel.

Figure 3 shows the measured magnetic field profile along the tube length. The maximum magnetic field is normalized to $1.0$.

Figure 3: Coil geometry and magnetic field map

The relation between the current in the coils surrounding the tube $I$ and the magnetic field along the electron beam trajectory $B$ is

PHYS 132 $B = \left(8.3 \times 10^{-5} ~\dfrac{\mathrm{T}}{\mathrm{A}}\right) \times I$. $(9a)$
PHYS 122 $B = \left(7.8 \times 10^{-4} ~\dfrac{\mathrm{T}}{\mathrm{A}}\right) \times I$. $(9b)$

A note on $L$

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.

Taking data


From Eq. (8), we see that we have two independent variables we can control – $V_A$ and $B$ – and one dependent variable that we can measure – $D$. Therefore, you can either…

  • …hold $V_A$ fixed while varying $B$
  • …hold $B$ fixed while varying $V_A$.

The Colab notebook provides two methods for using a constant $B$ field. The first requires you to measure the difference in the spot's position both with and without the magnetic field for each $V_A$. The second method instead lets this displacement be part of the fit equation.

Think about what advantages (or disadvantages) each method has and decide how you would like to take your data in order to test Eq. (8).

We provide a Google Colab notebook that you can use to plot and fit your data.

Physics 132: My voltage controls don't do anything!

You may have tripped the overcurrent protection feature of your power supply. See the video below for how to fix this.

Reminder: Using Logger Pro

Here's a reminder on how to make a plot and fit data in Logger Pro:

  1. Make a new file in Logger Pro.
  2. Enter your $x$-axis data in the first column.
  3. Enter your $y$-axis data in the second column.
  4. Double-click on a column header to rename it.
  5. Use the “Scale” button to resize your screen to show your data.
  6. Select the “Linear Fit” button to fit an equation to your line.
  7. Use the top menu “Data → New Data Set” to make additional plots.
    1. Click on a $y$-axis column and drag it to the plot area to display it

Submit your lab notebook

Make sure to submit your lab notebook by the end of the period. Download a copy of your notebook in PDF format and upload it to the appropriate spot on Canvas. Only one member of the group needs to submit to Canvas, but make sure everyone's name is on the document!

When you're finished, don't forget to log out of both Google and Canvas, and to close all browser windows before leaving!

Post-lab assignment


Answer the questions/prompts below in a new document (not your lab notebook) and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write the answers to these questions by yourself (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.

Conclusions

The conclusion is your interpretation and discussion of your data.

  • What do your data tell you?
  • How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.)
  • Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique?
  • Do your results lead to new questions?
  • Can you think of other ways to extend or improve the experiment?

In about one or two paragraphs, draw conclusions from the data you collected today. Address both the qualitative and quantitative aspects of the experiment and feel free to use plots, tables or anything else from your notebook to support your words. Don't include throw-away statements like “Looks good” or “Agrees pretty well”; instead, try to be precise.

REMINDER: Your post-lab assignment is due 24 hours after your lab concludes. Submit a single PDF on Canvas.