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RATIO OF ELECTRON CHARGE TO MASS (e/m) LAB

  INTRODUCTION  

J. J. Thomson, using a device similar to a cathode ray tube (CRT, see 2nd lab),

discovered the electron and measured the ratio of its electric charge (e) to its

mass (m). Thomson's experiment was concerned with observing the deflection of a beam

of particles in a combined electric and magnetic field. Its result established: 1)

the existence of the electron; 2) the fact that the electron has a mass (me); 3) the

fact that the electron has a charge (e); 4) that both charge and mass are quantized;

5) the ratio of e/m. In this lab we will repeat Thomson's measurement by observing

the deflection of an electron beam by a magnetic field B→.

This lab will allow you to experiment with the same type of equipment used for the

discovery of the electron. J. J. Thomson's discovery opened the door to our present

understanding of nature and to the technical world we are living in. The electrons

orbiting around nuclei are what make matter what it is, allow chemists to develop

new compounds, and allow biological phenomena to be explained. The electrons

oscillating in a radio antenna enable radios to transmit information. The controlled

flow of electrons through a semiconductor allows a computer to process data; and it

is a beam of electrons that allows specimens to be seen with an electron microscope.

It is with a beam of very high energy electrons that scientists have established

that nucleons are made up of components which are called quarks and gluons.

Possibly, these new components of matter will shape the way mankind will live 100

years from now in a way similar to that in which Thomson's discovery is responsible

for the way we are living today.

There are two basic physical phenomena which play a significant role in the

experiment carried out in this lab: the existence of a magnetic field associated

with an electric current, and the deflection of a moving charged particle in a

magnetic field. Before we discuss the experiment itself we will briefly review these

two phenomena, and will also discuss the fact that the earth has its own magnetic

field.

2

x

y

z

rdl

dB

I

Figure 1 Biot Savert Law

  MAGNETIC FIELDS AND FORCES  

The forces between electric currents are called

magnetic forces, because the same phenomenon

accounts for the forces acting between magnetic

materials, such as pieces of magnetized iron.

William Gilbert, Queen Elizabeth I's physician,

noted that a magnet has two poles at which magnetic

effects seem to be concentrated. He also showed

that like poles repel each other, whereas unlike

poles attract each other. Today we explain the

forces between magnetic poles in exact analogy with

the electrostatic forces between charges (lst lab)

by introducing a magnetic field B→ represented by

field lines. As we shall see later, the relation

between magnetic field lines and magnetic forces is

more complicated than in the case of electrostatic forces.

The fact that electric currents are intimately related to the magnetic properties of

materials was realized by Oersted, when he placed a magnetic needle over a wire and

observed that the needle moved when current flowed through the wire. Ampere then

observed that two conductors act upon one another: sometimes they attract and

sometimes they repel. It was Faraday who observed that an electrical current

produced a region of "magnetic force"; this demonstrated that the current element

acts as a source of magnetic fields (See Fig. 1). J.B. Biot and F. Savart were the

first to report to the French Academy the dependence of the "magnetic force" due to

a current flowing in a long wire upon the distance from the wire. The following

relation, known as the Biot-Savart Law, gives the field intensity dB→ observed at a

distance r→ from an element of current I dl

→:

Biot-Savart Law  d B→  = µo I 

dl→ x r

→ 

 r3   (1)

3

Figure 2 Right Hand Rule

The vector d B→ is tangential to the magnetic field

lines, which in the case of a straight wire conductor

are represented by concentric circles in a plane

perpendicular to the conductor. Figure 2 shows such a

B→ field and also illustrates (together with Fig. 5)

the "right hand rule", which gives the relation

between the directions of the 3 vectors B→, dl

→ and r

→.

If the thumb of the right hand is pointed along the

direction of the current I→, the fingers curl in the

direction B→. To increase the strength of B

→ in a

given volume, one usually uses a solenoid. In this

case the field B→ generated by each winding of the

coil will add up inside the solenoid, as shown in

Figure 3. The same figure also shows the similarity

between the fields of a solenoid and of a permanent

magnet. Notice that the magnetic field inside the

solenoid is nearly uniform. (This is true only when

the length of the solenoid is large compared to its

diameter). The solenoid is the classic device for

generating a magnetic field. It is the analog of the

parallel plate capacitor, which is the classic device

for generating a uniform electric field throughout a

given volume. Magnetic fields of a specific shape can

be generated by a system of coils.

Solenoid Permanent MagnetFigure 3

4

y

x

RR R

R

I I

Figure 4 Helmholtz Coil

Two parallel coils separated by a distance equal to the

radius of the coils (Figure 4 ) are known as Helmholtz

coils. They are frequently used because they generate a

magnetic field that is uniform over an appreciable

region about its midpoint. We will be using a system of

Helmholtz coils to carry out this lab's experiment. If

each one of the coils, with radius R, has N turns and

carries a current I, then the field at the center of

the system is,

Helmholtz Coils BC = .714 µo N IR

  (2)

The following table explains all of the symbols in

equation (2).

Quantity Units

[Bc] Tesla = 104 Gauss = NA m

mo (permeability constant) = 4 π x 10-7 Tesla mA

= 4 π x 10-7 V mA sec

[R] meters (m)

[I] amps (A)

[N] number of turns in coil

Let us now look at the force exerted by a magnetic field B→ on a moving charged

particle. It was the American physicist H. A. Rowland who first observed that a

particle with charge q and velocity v→ moving in a magnetic field B

→ will be

subjected to a force F→. The direction of this force is perpendicular to the

velocity v→ and to B

→. If a particle enters a volume with both an electric field E

→

and a magnetic field B→ then the total force on the particle is given by the Lorentz

law.

Lorentz Law   F→ = q E

→ + q ( v

→ x  B

→)  (3)

This is an extremely important relation, connecting mechanics (force F→) to

5

Figure 5 Right Hand Rule

Particle Orbit

Center

Guiding

F

F F

F

r = mv/qB

B-field is uniform into page

Figure 6 Motion of ChargedParticles in a Magnetic Field

electromagnetism (the fields E→ and B

→). The

magnetic force is the cross product of two

vectors. Figure 5 reminds you once more of

the "right-hand rule" which is defined for

positive charge. You must be familiar with

this rule in order to carry out this

experiment.

As mentioned in the introduction, in this

experiment we will observe the deflection of

electrons in a magnetic field. To simplify

the experiment we will choose the direction

of the electron beam ( v→) to be perpendicular

to B→. In this case the force acting on the

electrons (charge q) is simply,

| |F→mag = q | |v→ | |B→ or Fmag = qvB (4)

According to the right-hand rule, F→mag is

perpendicular to v→ and consequently the

electrons would move in a circle as shown

in Figure 6, if they had a positive

charge. Since they have a negative charge,

the electrons will actually rotate in the

opposite direction from that shown in the

figure. (Remember from mechanics: if a→ ⊥

v→ then it must be a circular motion). The

centrifugal force responsible for

describing a circular motion is given by:

Fcenter =mv2

r(5)

6

Observed Field Hypothetical Magnet Current Loop Inside Earth

Figure 7

By equating Egs. 3 and 4

Fcenter = Fmag(6)

mv2

r = qvB

we can calculate the radius of curvature of the circular motion of a charged

particle moving perpendicularly to a magnetic field:

 r =  mq  vB  (7)

This simple relation is the basic equation that we will use to carry out the

measurement of e/m. By measuring the radius of curvature r, of an electron beam of

known velocity v→, deflected by a magnetic field B

→, we can calculate the ratio q/m.

Before we describe how to do this experiment, there is one last topic to cover: the

earth's magnetic field! This field will have an effect on our electron beam, hence

we must properly take account of it - otherwise, our measurement of e/m will be

wrong.

  THE EARTH'S MAGNETIC FIELD  

The earth's magnetic field is the field of a magnetic dipole, which means that it is

equivalent to the external field of a huge bar magnet. The lines of force of such a

field are directed not towards the geographic poles but rather towards the magnetic

7

B

e-

Helmholtz Coils

Electron Gun

Glass Enclosurewith Low PressureMercury

Figure 8 e/m Experiment Setup - Main Aparatus

poles (The magnetic north

pole is located near the

geographic south pole.)

They are also directed

(except at the equator)

towards or away from the

center of the earth - as

shown in Figure 7.

The intensity of the field

at the surface is on the

order of one Gauss.

Sediments of magnetic

materials (iron, cobalt,

nickel) can drastically

change the local pattern of this field which has been carefully mapped, most

recently with the use of satellites. After centuries of research, the earth's

magnetic field remains one of the best described and least understood of all

planetary phenomena. The history of the earth's magnetic field has been traced back

3.6 million years, and it has been established that during this time the earth's

field has reversed nine times. To establish such a fact two elements were necessary:

the magnetic "memory" of volcanic rocks, together with the presence in the same

rocks of atomic clocks that begin to run just when their magnetism is acquired. The

memory elements themselves are magnetic "domains", tiny bodies in which magnetism is

uniform. These bodies consist of iron and titanium oxide. At temperatures above a

few hundred degrees (depending upon the chemical composition) these domains are

nonmagnetic. When a domain cools it becomes magnetized in the direction of the

surrounding magnetic field. The atomic clocks that record the time of the lava

solidification are based on the radioactive decay of potassium 40 into argon 40.

This decay (transformation of potassium into argon) takes place at a constant rate

similar to the decay rate of an RC circuit (see second lab). The argon is trapped

within the crystal structure of the minerals, and if the minerals are not heated or

changed in some way, it accumulates there. The amount of trapped argon is a function

of the amount of potassium present, and the length of time since the decay and

entrapment process began. The potassium-argon dating method has now been

successfully applied to rocks from nearly 100 magnetized volcanic formations, with

ages ranging from the present back to 3.6 million years; nine earth magnetic field

reversals were observed during this time. You should not worry about the earth's

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Accelerating Voltage

E

GridFilament

e

e Beam

Figure 9 Electron Gun

magnetic field changing during your experiment; the data of volcanic rocks shows

thaw it takes about 5000 years for a

field reversal to take place. You must,

however, be aware that there is an earth

magnetic field B→

E which affects this

experiment.

  EXPERIMENT 1  

Measure the ratio of the electron's

charge to the electron's mass in

Coulomb/kg.

As described above, the basic relation for this measurement is given by Eq. 7:

r = mq vB

(7)

This relation tells us that in carrying out the measurement of e/m we need three

basic elements:

1) a beam of electrons with known velocity v→.

2) a magnetic field (uniform over the region where the electrons will

describe a circular trajectory).

3) a way to observe the electron's path, so that we can measure the radius of

curvature.

Figure 8 shows a schematic drawing of the equipment used to measure e/m.

THE ELECTRON GUN

We generate a beam of electrons with an electron gun, very similar to the one

described in the 2nd lab for the CRT. It is shown schematically in Figure 9. The

electrons are accelerated to a final velocity v→, such that their kinetic energy is

equal to the work done by the accelerating potential V:

mv2

2 = qV

Electron velocity: v = (2 qm V)1/2 (8)

9

mAA V

e

0.5m

Glass Bulb

Axis of Rotation

Pins

Electron Gun

Electron Orbit

Power Supply

Helmholtz Coils

Field Filament

Anode

Figure 10 e/m Experiment Setup

B

B

E

C

Axis of Helmholtz Coils

Figure 11Alignment of e/m Apparatus withEarth's Magnetic Field

Figure 10 shows a view of the electrical connections for the equipment used.

The electron gun is controlled by 2 knobs. The "ANODE" knob, which sets the

accelerating potential, and the "FILAMENT" knob, which sets the current in the

filament, thereby controlling the

electron beam current Ianode.

THE MAGNETIC FIELD

The magnetic field is generated by

a set of Helmholtz coils. Formula

1 gives the value of the field at

the center of the system. The

number of turns for each coil is N

= 72. With a ruler, you can

measure the radius of the coils or

the distance between them, and

then calculate the value of B→.

The magnetic field is aligned

along the axis of the coil system

(see Figure 11), and its direction is determined by the "right-hand rule" (see Fig.

2). As mentioned previously, the earth also has a magnetic field, which cannot be

neglected in this experiment. Each apparatus

has been individually aligned, with the help

of a compass needle, in such a way that B→C

of the coil is in the same direction as B→E

of the earth (see Fig. 11).

Consequently, the magnetic field BT that

will deflect the electrons is:

B→T = B

→E ± B

→C (9)

where the ± sign depends upon the direction

of the current in the coil. The knob labeled

"FIELD", on the power supply (see Fig. 10),

will allow you to vary the strength of B→C,

1 0

while the meter above the knob gives a reading of the current I through both coils.

ELECTRON TRAJECTORY

Electrons are infinitesimally small objects (radius < 10-16cm) that cannot be seen

by the naked eye. In order to observe their trajectory without blocking their path,

the electron gun is installed in a glass enclosure (25 cm diameter) which contains

low pressure mercury (Hg) gas. The electrons (with 50 eV K.E.) will excite the Hg

atoms (requiring 2 eV) which then emit a blue light. The electron trajectory can be

observed (in a darkened room) as a ring of blue light. The glass bulb also contains,

along one of its diameters, a set of pins. The distances from the anode slit of the

electron gun are:

.065 m, .078 m, .090 m, .103 m and .115 m.

Some of these pins are still covered with a fluorescent material which emits light

when struck by the electron beam.

By varying the "FIELD" control knob you change the current I in the Helmholtz coils,

producing different values of B→C, and forcing the electrons to describe different

orbits. Certain values of B→C will allow the electron beam to strike the calibrated

pins. Knowing B→C and the radius of the electron beam will allow you to determine

e/m, provided the accelerating potential is known. The equations summarized below

are used in calculating e/m.

Eq.(7) r = mq vB

Eq.(8) v = (2 qm V)1/2

Eq.(9) B→T = B

→E ± B

→C

We can combine these equations and write,

B→C =

1r √2V mq ± B

→E

In this equation there are two unknowns: (q/m) and B→E. It will require a minimum of

two measurements (r1 and r2 for values of B→

Cl and B→C2) to determine both. The

measurement will be carried out by varying B→C, measuring r and plotting B

→C (

1r)

(i.e. B→C on the vertical axis and (

1r) on the horizontal). The data must fit a

1 1

straight lines with:

Intercept = ± BE

Slope = √2V mqor

em = 2V

(Slope)2(7)

Make a linear-least-squares fit to the data and calculate the half thickness. This

may be easily performed using MatLab or with some calculators. You can check out Dr

Erlenmeyer's Least-Squares fit web page and the corresponding Theory page for more

details. The measurement will be done for both orientations of BC: one with BT = BE

+ BC and one with BT = BE - BC

PROCEDURE

Measure the diameter of the Helmholtz coils with the meter stick. To calculate the

magnetic field of the center of the coils you need, according to Eq. 2, the radius

R. the number of windings N = 72, as seen in Fig. 4.2, and the current in

each coil.

BC = .714 (µo N / R) I (2)

Express this equation as BC = constant x I, and calculate the value of the constant

as well as its units, using mo = 4p x 10-7 Tesla meter/Amp.

The equipment is wired up according to the diagram in Figure l0.

(1) Turn both "FIELD" and "FILAMENT" controls fully counterclockwise, before

turning the power switch either ON or OFF.

(2) Switch the power supply ON.

(3) Turn the "ANODE" so that the meter directly above reads 50V anode voltage.

(4) Use the "FILAMENT" control to increase the filament current slowly, until

the center meter reads about .8 mA. This meter indicates the anode current

between the filament and anode; it should not exceed 1.0 mA. The anode current

may drift (as the equipment warms up) over the first 15 minutes, changing the

anode voltage. If necessary, reset the controls to keep the filament current at

.8 mA and the anode voltage at 50V. You are now ready to start the experiment

if the room is darkened. If the small lamp is properly placed on top of the

power supply, it illuminates the meters and also gives enough light for you to

1 2

write in your notebook. Stand on a lowered stool, if necessary, so that you can

look down into the tube. You should see the electron beam as a thin blue line

emerging from the anode slit. If necessary, rotate the tube on its axis so that

the electrons move towards your right as you look down. Increase the coil

current using the FIELD control and you should see the electron beam bend into

a circle. If the beam bends the wrong way, turn the FIELD control fully

counterclockwise, reducing the coil current to zero, and interchanse the lead

at the power supply connections. If the electron beam does not stay flat in a

circle, rotate the glass tube slightly on its axis until the electron beam

forms a flat circle.

Your equipment is now ready to take data.

* Vary the current in the Helmholtz coils (FIELD knob) until the outer edge of the

electron beam matches the outer edge of each pin in the tube.

* Record in your labbook the coil current settings for each measurement

corresponding to an electron orbit of radius r. The distances from the anode to

the pins (the diameter of the electron orbit) are

.065 m, .078 m, .090 m, .103 m, and .115 m.

* Calculate the field B→C using Eq. 2.

* Plot B→C as a function of 1/r.

* Rotate the glass tube 180° on its axis, reduce the coil current to zero, and

interchange the leads at the power supply. The electron beam now moves towards

your left as you look down. Obtain a new set of current values for matching the

outer edge of the electron beam to the outer edges of the pins in the tube. Make

a second plot of B→C as a function of 1/r. Your partner should repeat the above

measurements with an anode voltage of 60 volts.

Your writeup should include a plot of B→

C vs. (1/r) for both measurements. In

addition: Calculate e/m in Coul/kg for each slope measurement (beam clockwise and

counterclockwise), and also give the average value and the percentage error. Show

that the dimensions of the relation you are using to calculate e/m (Eq. 9) are

indeed Coul/kg. An important relation that allows you to relate units of mechanics,

such as kg, m, sec. to units of electricity-and magnetism, such as A, V, is the

following energy relation.

Joule = N m = Kg m2

sec2 = AVS = Coul V

1 3

* Calculate the earth's magnetic field B→E (in Gauss) at the location of your e/m

apparatus, this is given by the intercept of the line fitted through your data

points and the BC axis. Sketch for both measurements:

1) the direction of the electron beam.

2) the direction of the centripetal force and the Lorentz force acting on the

electron beam.

3) the direction of the current in the Helmholtz coils.

4) the direction of the field B→C.

Indicate if it was the measurement with (BE + BC) or with (BE - BC).

* The beam of electrons also represents a current and consequently must generate a

magnetic field B→

E. Does this field point in the same or in the opposite

direction as the field generated by the Helmholtz coils? Remember that an

electric current is defined for positive charge carriers.

  EXPERIMENT 2  

Each one of your setups has a permanent magnetic bar. One end of the bar has a

colored tape collar. By observing the motion of the electron beam in the magnetic

field of the permanent magnet, determine if the end with the colored collar is the

north or the south pole. Enter in your lab notebook the color of your permanent

magnet and its polarity. For A-2O, A-25 and A-90 students: Choose any point in the

vicinity of the bar magnet, and with the equipment you have, determine the intensity

of the field at this point.

Observe that the electron bean describes a spiral traiectory when the velocity

vector is not perpendicular to the field B→

C of the coils. (It may be useful to

reduce the velocity of the electrons). If the electron velocity

vector v is represented by a component parallel to B→C ( v

→||) and one component

perpendicular to B→C ( v

→⊥ ), which one of these two components will contribute to

the circular motion of the beam, and which one will simply describe a translational

motion?

1 4

How does the radius of the spiral motion depend upon the angle θ between v→

and B→C?

Sections of this writeup were taken from:

The Feynman Lectures on Physics, Vol. II, Feynman, Leighton, Sands/Addison-Wesley

Physics, J. Orear/MacMillan