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$e/m$ Of The Electron (PHYS142)


Last edited Jan-2023

For your final lab of the quarter you will make a measurement of the ratio of the electrical charge of the electron to its mass. The figure below shows the basic idea of the experiment which is an electron gun inside a vacuum tube which creates a beam of electrons that impact a phosphor screen to produce a glowing dot of light. When the electrons pass through an orthogonal magnetic field $\vec B$ they are deflected. If you know the energy of the electrons, the magnitude of $\vec B$, the length (L) of the electrons path through $\vec B$ and the deflection, you can calculate the ratio $\frac{e}{m}$.

From left to right the experimental apparatus consists of a source of electrons (Cathode) which are then accelerated along a line to the right by a potential difference (Vacc), this is referred to as the acceleration region. After leaving the acceleration region the electrons pass through a magnetic field created by a coil of wire which deflects their path.

In the next section we develop the relationship between the electrons deflection (d), $\vec B$, and $\frac{e}{m}$.

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),

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

where $\theta$ 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

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

Experimental setup


The apparatus (shown in photo above) consists of:

  • A Cathode Ray Tube (CRT) with a pair of rectangular coils built into the plastic shield.
  • A 500V power supply to power the CRT.
  • A 50V power supply to send a current through the rectangular coils built into the CRT shielding.
  • An iOLab device.
  • A couple of DMM's.
  • A couple of standard resistors which you may or may not need.

Lab Notebook Template and Plotting Script

Wiring the CRT

Note that it is possible to damage the CRT by wiring it up incorrectly. If you need to change any of the connections begin by Turning The Power Supply OFF. Then have your TA check the wiring Before turning the power supply back on.

The connections of the electron tube to the 500V power supply should already be made for you. Check to make sure that the connections are as shown in Fig. 2a. The left portion of the 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. Note that the schematic shows a DMM in the circuit, as an ammeter, for the coils that produce the magnetic field. This is only necessary if the current reading on the 50V power supply is not being given to enough precision. Depending on what accelerating voltages you use the use of this ammeter may not be necessary.

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

What do the connections inside the electron tube look like?

Figure 2b: Electron Tube wiring diagram showing internal components and connections to the power supply. Note that this diagram simply shows how the internal components of the electron tube relate to the connections you make between the power supply and the wires coming out of the back of the tube. We include it just in case you are curious, you do NOT need to make any of these connections.

Wiring photos & connections

The photo above shows the CRT connections to the high voltage power supply when it is correctly wired up without the DMM.

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.

Experimental Task

Your goal for this lab is to perform an experiment which measures the ratio of the electrons charge to its mass. Here are some tips to get you started.

  • The measurement is easier to make for higher accelerating voltages. Start by making the measurement for the largest Vacc available to you, which should be something around 500V. Repeat the measurement for a second, lower accelerating voltage.
  • Don't forget that you can easily reverse the direction of the applied $\vec B$ and thus measure both positive and negative deflections.
  • Don't forget that there are other B fields present in the lab and these could have an impact on your measurements.
  • Do not try to calculate the value of $\vec B$ as a function of current to the coils. The coils are not in a Helmholtz configuration so the calculation is difficult. Instead figure out how to make measurements which will allow you to parameterize the relationship between coil current and $\vec B$.

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 result of a measurement of the 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 approximately $B = \left(8.3 \times 10^{-5} ~\dfrac{\mathrm{T}}{\mathrm{A}}\right) \times I$. Approximately means that this conversion was measured for one particular set of coils. Your coils should be within a factor of about 2 of this value which is provided as a sanity check.

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.

In the lab there is a sample of the structure inside the CRT which you can use to help visualize the proper drift region and estimate it's length.

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!

Report: Summary and Conclusions


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!

Your individual summary and conclusions are due 48 hours after the end of the lab.