2024 notes

  • Took only about 1 hr if things were set up ahead of time
  • Most folks seemed to not need to wire stuff up, TAs might've done it in some cases?
  • Asking about the trajectory was interesting; useful stuff.
  • Some groups using small deflecting voltages had issues
  • When groups didn't include offset, didn't fit to data.
    • Adding an offset to the fit would delay issue until part 2
    • TA Idea: add a “zero deflection” point that people subtract off as an explicit instruction

Electric Fields II: Acceleration and Deflection of Electrons

In Part I, we observed the electric fields (and resulting electrostatic potentials) produced by different conductor configurations. In Part II, we will use parallel plate conductors to produce uniform electric fields that can be used to move electrons around – either accelerating an electron moving in the same direction as the field or deflecting an electron moving perpendicular to the field.

Introduction


Cathode ray tube (CRT)

The cathode ray tube (CRT) is a vacuum tube consisting of the following three basic components which are shown in Fig. 1:

  • electron gun,
  • electrostatic deflection system, and
  • fluorescent screen.
Figure 1: Schematic of 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 plates are flared to permit wide angles of beam deflection. Finally, there is a screen which fluoresces when struck by the energetic electron beam.

The cathode ray tube of Fig. 1 is the basic component of analog oscilloscopes and older-style TV picture tubes.

Acceleration of electrons

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

$\frac{1}{2}m v_l^2 = e V_a$, $(1)$

where we have used the term $v_l$ to designate that in a cathode ray tube this velocity is longitudinal, i.e. along the length of the tube.

Deflection of the electron beam

The internal geometry of our cathode ray tube is schematically shown in Fig. 2.

Figure 2: Electron beam path geometry in a cathode ray tube

If $V_d$ is the potential difference between one set of deflecting plates and $d$ the distance between them, then the transverse electrostatic field in that region is $E = V_d/d$. Thus, the transverse acceleration during the time that the electron is inside the deflecting region is

$a = \dfrac{F}{m} = \dfrac{e V_d}{m d}$. $(2)$

This means that a transverse velocity

$v_t = \left(\dfrac{e V_d}{m d}\right)\Delta t$ $(3)$

is acquired, where $\Delta t$ was obtained from the longitudinal velocity $v_l$ and the length of the deflection region $\ell$. The electron will then drift over a distance of length $L$ resulting in a transverse displacement from the direct beam direction of $D$.

A measure of the deflection angle is the ratio of transverse to longitudinal velocity

$\dfrac{v_t}{v_l} = \tan(\theta) = \dfrac{eV_d}{m d}\dfrac{\ell}{v_l^2}$. $(4)$

However, from Eq. (3), $v_l^2$ can be eliminated, so

$\tan(\theta) = \dfrac{V_d}{V_a}\dfrac{\ell}{2d}$. $(5)$

After the electron leaves the deflection region, the electrostatic forces are zero, and therefore the electron continues to move in a straight line. As a result, we expect to observe a total deflection $D$ of the electron beam on the screen, given by

$D = L \tan(\theta) = \dfrac{L\ell}{2d}\dfrac{V_d}{V_a} = k\dfrac{V_d}{V_a}$, $(6)$

where we simplify the equation by introducing the constant $k = L\ell/2d$ which depends only on the tube geometry. Thus, the deflection of an electron beam should be proportional to deflecting voltage and inversely proportional to accelerating voltage.

Note for instructors:


Info from the Standard Handbook of Video and Television Engineering section 5.5.2a

The expected vertical deflection from a set of non-parallel plates is

$ y = \dfrac{d_lV_d}{2V_ba_1}$ $ \dfrac{\ln\left(\dfrac{a_2}{a_1}\right)}{\dfrac{a_2}{a_1}-1}$

where:

  • $d_l$ is the length of the plates
  • $V_d$ is the deflecting voltage
  • $V_b$ is the beam acceleration voltage
  • $a_1$ is the entrance separation
  • $a_2$ is the exit separation

For our setup, this means our correction factor is around $ \dfrac{\ln\left(\dfrac{a_2}{a_1}\right)}{\dfrac{a_2}{a_1}-1} $

for Vertical plates,

$\ln(.978/.381) / (.978/.381 -1 ) \approx 0.61 $

for Horizontal plates,

$\ln(1.15/.434) / (1.15/.434 -1 ) \approx 0.59 $

Long story short, count only 60% of the distance of the flared plates.

Experimental Procedure


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.

Apparatus

A schematic of the CRT showing the relevant dimensions is shown below in Fig. 3.

CAUTION: An exposed cathode ray tube is dangerous to handle even if no voltages are applied. Since the tube is evacuated, atmospheric pressure exerts a large force on the glass envelope which could cause a violent implosion. To protect you, the tube is surrounded by a plastic cover. Leave the cover in place at all times.

CAUTION: For this experiment we shall operate the tube at about 500 volts. There is a shock hazard, so - be careful - do not touch any bare wires while the power supply is on.

Figure 3: 3BP1-A Tube Details

The electrical connections between the cathode ray tube and the power supplies are shown in Fig. 4. Note that the power supply wiring diagram (Figs. 4b and 4c) differs depending on which model power supply you are using. Check your setup and use the appropriate diagram.


Figure 4a: Voltages applied to electron tube

Figure 4b: Wiring for 3B Scientific power supply (Room 107 only). The orange and teal lines indicate points that should be connected to the same set of wires. DMM indicates a digital multimeter.

Figure 4c: Wiring for Heathkit power supply, model IP-17 (Room 109 only). The orange and teal lines indicate points that should be connected to the same set of wires. DMM indicates a digital multimeter.

Figure 4d: 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.

Figure 4e: Photo of the twin leads from the electron tube. Each controls one of the deflection plates.

Figure 4f: Photo of the $\pm24\text{ V}$ supply. Make sure you are using the variable output and that the switch is set to $24\text{ V.}$

Finding the spot

Turn on the CRT by switching on the high voltage power supply and adjusting the voltage up to approximately 500 V. You should find a spot somewhere on the CRT face. Adjust the focus control at the rear of the tube base to achieve the smallest spot possible.

Example of CRT Spot

An example of the green CRT spot. You may find your CRT produces a diagonal line instead; as long as you can measure from a consistent location you should be fine.

Loosen the clamp which holds the tube assembly to the lab bench. Remove any magnets which may be attached to the tube assembly. Change the orientation of the apparatus by rotating the tube base on the bench.

Reduce the accelerating voltage $V_a$ to a minimum value which still maintains a visible spot on the screen. Rotate the tube on the bench again.

The spot's position is affected by the earth's magnetic field which also exerts a force on moving electrons. Most commercial devices that use CRTs such as analog oscilloscopes are magnetically shielded to avoid this problem.

Re-clamp the tube assembly on the bench to avoid an inadvertent shift of the spot during the experiment.

Initial observations

Describe what you see as you vary the parameters you can control. At a minimum, this should include varying the accelerating voltage $V_A$, the deflecting voltage $V_D$, and the orientation of the apparatus.

Record these observations in your lab notebook.

$D$ vs. $V_d$ for vertical deflection plates

Using the $\pm 24\text{ V}$ supply, apply both positive and negative voltages across the vertical deflection plates. If the beam moves horizontally, exchange the twin leads so that the beam moves vertically. Before taking data, scan through the full range of beam deflections to check for dead spots on the CRT screen. If such a spot is encountered, it will be necessary to shift the beam slightly by placing a small magnet on top of the tube housing.

CAUTION: If a magnet is necessary, be sure not to move it during the experiment.

Make measurements of the spot deflection, $D$ (displacement from where $V_d= 0$) as function of the applied deflecting voltage $V_d$ and record them in your notebook.

According to Eq. (6), we expect the spot deflection to increase linearly with increasing deflection voltage (for a fixed value of accelerating voltage). Let's plot our data to see if it is so! We provide a Google Colab notebook to help with data visualization and fitting below.

Use the notebook above to study a single data set first ($D$ vs. $V_d$ for one value of $V_a$), then repeat your measurements of $D$ vs. $V_d$ for two more values of the accelerating voltage $V_a$.

Are your data in agreement with Eq. (6)? Do the slopes agree with your expectations?

What is the relationship between $D$ and $V_a$?

Next, we want to test the relationship between $D$ and $V_a$.

Chose a value of the deflecting voltage for which you measured $D$ in each of your data sets above. For each accelerating voltage, note down the displacement at that deflecting voltage. (Put another way, we are finding one pair of $V_a$ and $D$ values at fixed value of $V_d$ from each data set above.)

Return to the Google Colab notebook and make a new plot of $D$ as a function of $1/V_a$. Are your data consistent with Eq. (6)? Is your slope in agreement with expectations?

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.