# PHY 124 Lab 8 - Measurement of e/m for the electron

The purpose of this laboratory is the measurement of the charge (e) over mass (m) ratio e/m for the electron and to study qualitatively the motion of charged particles in a magnetic field.

If you need the .pdf version of these instructions you can get them here.

## Equipment

• e/m apparatus (two similar, but non-identical ones, from different manufacturers; we call them “A” and “B”)
• meter stick
• black cloth

There are kinds of 2 e/m apparatus. Apparatus A has built-in power supplies for delivering the magnet current and the accelerating voltage. Apparatus B uses external power supplies for these tasks. The settings you use for the two apparatus will be slightly different. Please make sure you identify correctly which one is “your apparatus” in the lab and use the settings and instructions that apply to it.

## Background to the experiment

The principle of the e/m measurement is as follows (see the figures above and below): electrons “boiled” off a heated metal cathode are accelerated by an accelerating voltage difference V applied between the negative cathode and an anode kept near ground potential. The anode has a small hole in it that allows a small fraction of the electrons to pass through, thereby creating an electron beam with energy kinetic energy $KE = eV$ inside the glass tube mounted on top of the power supply box. (Recall that $e$ is the elementary unit of charge: $1.602 \times 10^{-19}$ Coulomb.) A small amount of helium gas is present inside the tube, and when forced onto a circular path by the magnetic field [See Ch18 sheet 7 in the online lecture notes for PHY 124. Or see Sec. 24.5, “Magnetic Fields Exert Forces on Moving Charges”, pp. 789-795, in Knight, Jones, and Field, COLLEGE PHYSICS: A Strategic Approach, 2nd ed. (KJF2), the optional textbook for PHY 124. A copy is bolted to a table in the help room, A-131 physics building, and a few copies are on closed reserve in the Math/Physics Library on the “C” floor of the physics building.] supplied by current flowing in the two “Helmholtz coils” shown in the figure, some of the electrons excite helium atoms in collisions. The excited helium atoms emit light that we see as the blue-green glow in the color pictures in this lab manual. The apparatus allows one to determine – albeit with non-negligible experimental uncertainty – the diameter of the circular path of the electron beam. (The details of the procedure for doing this depends on which apparatus you are using; see below.) This diameter depends on the energy of the electron beam and on the strength of the magnetic field B inside the spherical vacuum tube. The B field is approximately perpendicular to the plane of the Helmholtz coils, and, when the tube is properly oriented, the circular path of the electrons lies in the midplane between the coils. The figure below will help you to use the right-hand-rule (remember the electrons have negative charge!) to visualize the directions of the electron velocity $v$, the magnetic field $B$, and the (centripetal) force $F$.

In the MapleTA online pretest for Lab 8, you must derive the e/m relation you will use in your analysis for this lab. The kinetic energy $KE$ and hence the velocity $v$ of the electrons are determined by the accelerating voltage they have traversed (see Ch16 sheet 25). The radius $r$ of the circular path can be related to the electron velocity and magnetic field (see Ch18 sheet 8,9). By combining these equations you will end up with a formula that relates e/m to the quantities you measure in this lab.

Use the equations on Ch16 sheet 25 and the relation $KE=\frac{1}{2}mv^{2}$, solving for the velocity to get a relation between the electron velocity $v$ to the accelerating voltage $V$. Then use this relation and Eq. (18.2) [see Ch18 sheets 8,9] for the radius $r$ of the circular path of the electron. Substitute the equation for velocity into the equation for $r$ and then solve for e/m. This will relate e/m to the accelerating voltage $V$ and the magnetic field $B$. You should get the relation

$\Large \frac{e}{m}=\frac{2V}{B^{2}r^{2}}$
(8.1)

In this experiment you don't use the current flowing in a single loop of wire to generate the magnetic field $B$ that deflects the electrons. Instead, you use so-called “Helmholtz coils”, which is a special arrangement of two, identical multi-turn coils of wire that produces a $B$ field that is quite uniform near its geometrical center between the coils. The “Helmholtz condition” is that multi-turn coils are parallel, have identical currents $I$ flowing in the same direction, have the same radius $a$, and are separated by $a$. The formula (SI units) for the magnetic field produced near the geometric center between the Helmholtz coils – which is where the circular path of the electron beam is arranged to be – is

$\Large B= \left( \frac{4}{5} \right)^{3/2} \left( \frac{\mu_0 NI}{a} \right)$
(8.2)

where
$N$ = the number of turns in each coil of the Helmholtz pair = 130
$a$ = the (mid-)radius of the each coil in meters (see the figure above)
$\mu_0 = 4\pi \times 10^{-7}$ Tm/A is the permeability of free space [in terms of the magnetic constant $k^{\prime}$ used in the online lecture notes, $\mu_0 = 4 \pi k^{\prime}$]
$I$ = the current in amperes flowing in each coil.

Substituting Eq. (8.2) for the magnetic field, $B$, in Eq. (8.1) gives, for e/m,

$\Large \frac{e}{m}= \left( \frac{125}{32} \right) \left(\frac{a^2}{\mu_0^2N^2} \right) \left( \frac{V}{I^2r^2} \right) = \left( \frac{5}{4} \right)^{3} \frac{2Va^{2}}{(4\pi k'NIr)^2}$
(8.3)

You will use Eq. (8.3) [either in terms of $\mu_0$ or in terms of $k^{\prime}$] for the the quantitative work in Part II.

BEWARE: In this lab you will be measuring two radii, the radius of the circular path of the electron and the radius of the Helmholtz coils. The radius of the electron path will be referred to by the symbol $r$ and the radius of each (Helmholtz) coil (equal to the separation between them) will be referred to by the symbol $a$.

## Part I: Qualitative Exploration

Before you use the equipment to determine e/m, you will first investigate the relation between the voltage and current controls of the apparatus and the radius of the path of the electron. In this part you explore how the accelerating voltage $V$ and the magnet current $I$ influence the radius $r$ of the electron orbit. It is important to note that the voltage and current controls affect two different parts of the set up. The accelerating voltage, $V$, affects the electron beam and the magnet current, $I$, affects the magnetic field $B$ produced by the Helmholtz coils.

### Procedure

Apparatus A Apparatus B
Plug in the apparatus and turn the power on. Wait for 30 seconds during which the power supply runs a self-test. The power supply has dials (see the figure above) for the accelerating voltage $V$ (on the left) and the magnet current $I$ on the right. Set $V$ to ~150 V and $I$ to ~ 1.2 A. The power supplies should already be connected before you begin. If not, ask your TA to do it for you. Plug in the power supplies and turn the power on for both units. Use the 500 V adjust dial on the high voltage power supply to control the accelerating voltage $V$ and the current adjust on the e/m apparatus to control the current magnet current $I$. Set V to ~250 V and I to ~1.5 A.

You should see the electron circular path as a “blue-green” glow. This glow comes from the ionization of a small amount of Helium left in the vacuum tube by the electron beam. Note that you will probably have to either cover the coil with a dark cloth or dim the room lights to see the glow.

Keeping $V$ constant at either 150 V or 250 V, depending on your setup, increase $I$ in several steps and observe the radius $r$ of the electron path. Explain your observation in terms of the coil current $I$, the electron velocity $v$, and the relation (18.2) on Ch18 sheet 9.

Keeping $I$ constant at either ~1.2 or ~1.5 A, depending on your setup, increase $V$ in several steps and observe $r$. Explain your observation in terms of the accelerating voltage $V$, the electron velocity $v$, and the relation (18.2) on Ch18 sheet 9.

## Part II: Measurement of e/m

In order to get an accurate coil (mid-)radius, $a$, you will take several measurements and use their average for future calculations. Measure the vertical (from top to bottom) and the horizontal (from left to right) diameter (from the middle of the coil winding) for both coils and enter them on your worksheet . Notice that these values are referred to as $2a$ because they are diameters. Take the average of these four measurements to calculate the coil radius you will use in the analysis:

$\large a=\frac{1}{2}(2a_{v1}+2a_{v2}+2a_{h1}+2a_{h2})/4$

Estimate the uncertainty of one diameter and assume it is the same for each of the four measurements. Calculate the uncertainty for the average value of $a$ using Eqs.(E.5b) and (E.5a) in Uncertainty, Errors and Graphs.

### Procedure

Apparatus A Apparatus B
To measure the electron beam radius, $r$, you will use the marks etched on the glass scale in the vacuum tube. Be careful to notice that these marks are in cm and give the diameter of the beam path! You will take your data by keeping a fixed $I$ and varying $V$. Keep $I$ fixed at ~ 2 A and start with $V$ ~ 150 V. Increase $V$ in ~6 steps such that at each step the electron beam hits the cm marks etched on the glass scale in the center of the vacuum tube. It is more important that the beam hits these marks than to have equal $V$ steps. For each step record $V$, $I$, $r$ and your estimate of the uncertainty $\Delta r$ in the table on your worksheet.
On apparatus B the globe can be rotated, altering the direction of the beam path relative to the magnetic field. Before you begin to make this measurement you should ensure it is oriented so that the beam forms a circle and not a spiral. To measure the electron beam radius, $r$, you will use the illuminated mirrored scale on the back of the apparatus and the ruler in front of the apparatus. The mirrored scale helps to reduce parallax error, you do this by moving your head until the edge of the reflection of the beam in the mirror aligns with the edge of the actual beam. (This means that you are looking straight on at beam.) Then position the metal triangle on the front ruler so it touch the sides of the beam path and measure the beam path radius from the scale on the mirrored ruler. Take an average of your measurement of the radius when you are looking from the left and from the right as the radius of the beam. You will take your data by keeping a fixed $I$ and varying $V$. Keep $I$ fixed at ~2 A and start with $V$ ~200 V. Step $V$ up in 6 steps of ~50 V. For each step record $V$, $I$, $r$ and your estimate of the uncertainty $\Delta r$ . For each step record $V$, $I$, $r$ and your estimate of the error $\Delta r$ in the table on your worksheet.

When estimating $\Delta r$ for either apparatus, consider things like the fuzziness or the brightness of the beam. Note that $\Delta r$ does not have to be the same for each measurement.

Assume that the errors of $V$ and $I$ are negligible. Solve Eq. (8.3) for $V$ and write it such that it has the form $V=constant\times \frac{e}{m}\times (Ir)^2$ where “constant” stands for a factor containing only quantities that are constant during the experiment, including the Helmholtz coil radius $a$. Calculate the value of “constant” and enter it on your worksheet. Notice it includes the radius $a$ of the Helmholtz coils, which has an uncertainty so you must propagate the uncertainty of $a$ into the uncertainty of the constant using Eqs. (E.1) and (E.8) in Uncertainty, Errors and Graphs.

Transfer your recorded data into the form below.

V1  V   I1  A   r1  m   Δr1  m
V2  V   I2  A   r2  m   Δr2  m
V3  V   I3  A   r3  m   Δr3  m
V4  V   I4  A   r4  m   Δr4  m
V5  V   I5  A   r5  m   Δr5  m
V6  V   I6  A   r6  m   Δr6  m

When you click submit the computer calculates the quantity $(Ir)^2$ for each data point and then plots $V$ vs. $(Ir)^2$. Why do you plot $V$ vs $(Ir)^2$ and not $V$ vs $(Ir)$? Since you are considering the uncertainties in $V$ and $I$ to be negligible, all the uncerainty of your plot will come from the uncertainty in $r$, so you will only have horizontal error bars. The computer obtains the values for the error in $(Ir)^2$ by propagating the uncertainty in $r$ into the uncertainty of the quantity $(Ir)^2$ using Eqs. (E.1) and (E.8) in Uncertainty, Errors and Graphs. Write the slope of the graph and its uncertainty on your worksheet.

Next, set up an equation that relates the slope, $s$, to the “constant” and to e/m. Solve the equation for the ratio e/m. Using your uncertainties for $s$ and the “constant” calculate the uncertainty in e/m using Eq. (E.7) in Uncertainty, Errors and Graphs. Make sure you simplify units as much as possible. Compare your value to the established value that you can obtain by dividing the charge on the electron, $e=1.602\times10^{-19} C$ by its mass $m_{e}=9.109\times10^{-31} kg$