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Last edited Feb-2025

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 (Faraday's Law) which deflects their path.

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

The force on a charge moving in a magnetic field is

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),

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

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

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

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

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

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

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.

The connections of the electron tube to the 500V power supply are 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 double banana plug with the twisted yellow wires supplies current to the filament inside the CRT which is the source of electrons. This connector must be plugged into the 6V AC jacks on the left side of the power supply. Do NOT plug it into the 12V AC jacks as doing so will burn out the filament.
• The connection between the 6V AC filament and the minus side of the high voltage is necessary so that the electrons are immediately accelerated towards the positively charged end of the accelerating region.
• The center jack on the high voltage side of the power supply is electrical ground. For safety we connect the positive HV to ground so that the outside of the CRT remains at ground potential for safety.
• The pair of twin banana connectors with the brown cabling provide voltage to a pair of deflection plates. One connector for the vertical deflection plates and the other for the horizontal deflection plates. These deflection plates are used to steer the electron beam as was once done for oscilloscopes. For this lab we do not want them to affect the electrons as they travel through the acceleration region so we connect them to the positive side of the HV.
• The single banana wires with red/white brown(black)/white twisted wires apply the high voltage to the two ends of the CRT. The banana with the red and white twisted wire connects to the positive side of the HV and the one with the black and white twisted wire connects to the negative side of the HV. This creates the accelerating potential for the electrons.

The right side of the figure 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 this ammeter may not be necessary.

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

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.

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 around 500V.
• 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 $\vec B$.

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.

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.

For your individual summary discuss the following.

Your individual summary is due 48 hours after lab.

Congratulations! You have finished the experimental component of PHYS142.