Electron e/m
Skills and Discovery Laboratory
Full experiments – Measuring e/m for the electron
Learning Outcomes (what skills you will acquire)
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•
•
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Using a fine electron beam tube
o Practice at calibrating equipment
Propagation of uncertainties in measured quantities
Straight-line graph analysis; comparing a quantity measured with two different
techniques
Understanding how the motion of charged particles is influenced by a constant
magnetic field
Measuring the charge to mass ratio for tangential and radial electron beams
Preparatory Task:
Read the entire script; familiarise yourself with the learning outcomes and skills
required. If you are not confident in having acquired the skills you will be using today,
make use of the web resource where you can gain further practice. Write a short
paragraph in your lab book giving details about the experiment, its aims and a brief
outline of the theory behind the experiment.
Classical mechanics?
The derivation in the script is based on classical mechanics. Why are we justified in ignoring
special relativity for this calculation?
Given that a typical anode voltage is 200 volts calculate the speed of the electrons in the
beam (use Tipler to find any fundamental constants you need). Explain whether it is
reasonable to ignore special relativity in calculating the electrons' trajectories. [1 Mark]
1. What it’s about
Historical importance of this experiment.
In 1897 J. J. Thomson performed a series of wide-ranging experiments with farreaching consequences (the "discovery of the electron"). These results were crucial to the
development of the understanding of the electrical properties of matter. His experiments
confirmed that the speed of the electrons is significantly less than that of electromagnetic
waves, c; and gave an (e/m) of about 2000 times larger than that of a Hydrogen atom ionised
in an electrolysis experiment. It was not obvious at the time whether the large charge to mass
ratio was a consequence of a small mass, with the same unit of charge that was instrumental
in electrolysis, or whether it was owing to a large intrinsic charge. What he was able to show
was that this new particle, with its characteristic charge-to- mass ratio was a constituent of
every material that he was able to utilise as a cathode. This independence showed that the
particles of the beam are a common constituent of matter, which we now call the electron.
Technological importance of this experiment.
Although LCDs and plasma displays are being introduced almost all TVs in use today
rely on a device known as the cathode ray tube, or CRT, to display their images. The fine
beam tube you will use to perform this experiment is an elementary CRT. The manipulation
of the trajectory of an electron beam by applying (time-dependent) electromagnetic forces
which you will demonstrate is a key ingredient for displaying information on a TV screen.
Electron e/m
Skills and Discovery Laboratory
The principle that a charged particle's trajectory in uniform electric and magnetic
fields is dependent on its mass is used in a device called a mass spectrometer.
2. Preparation
2.1 Principle of the fine beam tube
Electrons are ejected from a heated cathode into a partially evacuated tube. Collisions
between the electrons and residual gas atoms or molecules yield positive ions along the path
of the electron beam. The ions serve to partially focus the beam, owing to the Coulomb
attraction which prevents the scattered electrons from straying far from the beam axis.
Collisions between electrons in the beam and gas atoms in the tube excite the latter, which
subsequently decay by emitting light. This is how we can see the trajectories of the electrons
in the beam.
Figure 1: a schematic of the fine beam tube.
The tube can provide either a radial or a tangential beam, the former directed
horizontally through the tube and the latter directed vertically. It is possible to switch
between the two beams by changing the electrical connections, as will be discussed later.
2.2 Electromagnetic Deflection of the Electron Beam
A magnetic field perpendicular to the axis of the tube can be applied by means of a
pair of Helmholtz coils. Each coil comprises N turns of copper wire through which a current
I is caused to flow by means of a low voltage supply. When the coils are connected correctly
and placed in the appropriate place, the magnetic induction B at the centre of the fine beam is:
8 µ 0 NI
,
(1)
5 5 a
where a is the coil radius. For the coils used in this experiment the parameters are N = 320
and a = 6.8 cm ( µ0 = 4π ×10−7 henry m −1 ).
B=
The force, F, on the electron is the Lorentz force, which is perpendicular to its
velocity and to the magnetic field and has magnitude:
F = Bev,
(2)
Electron e/m
Skills and Discovery Laboratory
where e is the magnitude of the electron charge and v its speed. When the magnetic field is
applied, the electrons follow a circular path. It is well known that a particle following a
v2
circular orbit must have an acceleration of magnitude
pointing towards the centre of the
R
circle. Therefore for an electron of mass m in an orbit of radius R the force acting on it must
have magnitude:
mv 2
F=
.
(3)
R
From equations (2) and (3) it follows that the charge to mass ratio, (e/m), for the electron is
given by:
e
v
=
.
(4)
m BR
Conservation of energy allows us to calculate the speed of the electrons in the beam. The
high voltage, V, supplied to the anode of the fine beam tube is related to the speed, v, thus:
1
eV = mv2 ,
(5)
2
or
2eV
v=
.
(6)
m
Finally,
e
2V
= 2 2,
(7)
m B R
where all the quantities on the right hand side can be measured.
3. Ready to Start
First, ensure that the unit supplying the high tension supply is switched off. Then,
connect the circuit as shown in the diagram. The anode voltage must always be reduced to
zero before switching on the cathode heater supply or when changing between the
horizontal and vertical beams . Allow two minutes for the cathode of the tangential beam to
warm up, then increase the anode voltage to between 100V and 200V. The electron beam
should be visible - a black cloth behind the tube improves visibility.
Figure 2 - electrical connections to run the beam.
Electron e/m
Skills and Discovery Laboratory
4. Experiment
Part A - the tangential beam
Now slowly increase the low tension supply to the Helmholtz coils. If the
connections have been made correctly, the electron beam will become circular, its radius
decreasing with increasing magnetic field strength. If you can not obtain a circular beam ask
for assistance from the demonstrator.
Task 1: direction of magnetic field
Deduce the direction of the magnetic field, given that the vector force F on the electron, of
charge -e, is given by F = −e( v × B) [1 Mark]
Choose an anode voltage and a current through the Helmholtz coil. Measure the
diameter of the electron beam. Tolerable precision is attained using vernier calipers, with a
mirror to reduce parallax errors.
Get your lab partner to measure independently the radius of the same beam. Then
both of you should take another measurement each, such that you have (at least) four
measurements of the diameter of the beam. Is the beam circular? Does this influence your
measurement strategy?
Task 2: repeated measurements of the diameter
Calculate the mean, standard deviation and standard error of your measurements of the
diameter.
Comment on any differences between the standard error for the distribution of results you
obtained and the precision of a single result (the precision of the instrument). Explain
clearly the implication this result has for your measurements of the diameter. [1 Mark]
Now with the same anode voltage change the current through the Helmholtz coils.
Take at least four measurements of the beam diameter. Repeat for a third value of Helmholtz
current for this anode voltage. Then repeat this measurement cycle for two other anode
voltages. You should have nine pairs of (anode voltage, Helmholtz current) settings, and a
mean value of the diameter and its uncertainty. The easiest way to analyse this data set is
graphically.
Bear in mind that you measured the diameter, not the radius, of the beam!
Equation (7) can be rearranged to be of the form
B 2 R2
1
=
V
e
2
m
( y = mx )
( )
i.e. a straight line. Tabulate values of the quantity
(8)
( B R 2 ) and its associated uncertainty.
2
2
Plot a graph and obtain the best-fit straight line. Deduce the slope of the line and its
uncertainty. Use these values to deduce (e/m) and its uncertainty.
Task 3: reporting a value for (e/m)
Based on your data what to you deduce for the electron's charge to mass ratio (e/m), and an
associated uncertainty? How does this compare with the accepted value? [1 Mark]
Electron e/m
Skills and Discovery Laboratory
Part B - the radial beam
Reduce the anode voltage to zero. Switch to the horizontal beam (ask the
demonstrator for assistance if you are unsure how to do this) and allow two minutes for the
cathode to warm up. Increase the anode voltage to between 100V and 200V. With zero
current through the Helmholtz coils, the beam should travel horizontally through the tube,
producing a spot on the luminescent screen. If a current is now passed through the coils, the
beam will be deflected as shown in the diagram. The opposite sense of deflection may be
obtained by reversing the polarity of the voltage supplied to the Helmholtz coils.
Figure 3. The geometry of the deflection of a radial electron beam subject to a horizontal
magnetic field.
We first need to determine the radius of curvature of the beam. When the spot attains
the edge of the luminescent screen, the corresponding value of l is given by 2 l = 9.5 cm.
Use a ruler to determine the distance s of the electron gun from the centre 0 of the tube (in
practice, measure p − s and s + p, where p = 6.4 cm is the tube radius). The radius of
curvature of the beam, R, is given by Pythagoras’ theorem as:
R=
d 2 + L2
,
2d
(9)
where
d = p sin θ ,
L = s + p cosθ ,
(10)
(11)
and
θ=
l
p
(12)
when expressed in radians.
Task 4: calculating the radius of curvature R of the electron beam
Based on your data, what to you deduce for the value for the electron beam's radius of
curvature, R, and its associated uncertainty? [1 Mark]
Note the current through the Helmholtz coils and the anode voltage. You might wish
to use the digital multimeter to measure the current. Repeat for different value of the anode
voltage.
Electron e/m
Skills and Discovery Laboratory
Equation (7) can be rearranged to be of the form
B2 =
2
R2 e
( m)
V
( y = mx )
(13)
i.e. a straight line. Tabulate values of the quantity ( B 2 ) and its associated uncertainty. Plot a
graph of B2 on the y-axis against V on the x-axis, and obtain the best- fit straight line. Deduce
the slope of the line and its uncertainty. Use these values and your estimate for the beam's
radius of curvature, R, and its uncertainty, to deduce (e/m) and its uncertainty.
Task 5: reporting a value for (e/m)
Based on your data what to you deduce for the value for the electron's charge to mass ratio
(e/m), and an associated uncertainty? How does this compare with the accepted value and
the value you deduced in part A? [1 Mark]
5. Exploration: If you have time
Think about the limiting errors in this experiment. Can you think of ways of
improving the experiment. There are other students performing this experiment at the same
time as you. Can you incorporate their measurements of (e/m) with your own to obtain a
class value for this fundamental ratio? Thomson's result (1897) was 30% inaccurate - how is
(e/m) measured with better precision now?
6. To conclude
At the time of Thomson's original experiment the results were interpreted as evidence
for the particle nature of electrons; disproving the then topical wave theory preferred by
many. J. J. Thomson won a Nobel Prize in 1906 "in recognition of the great merits of his
theoretical and experimental investigations on the conduction of electricity by gases". J. J.
Thomson's son, G. P. Thomson, demonstrated diffraction with electrons, i.e. confirming they
can exhibit wave behaviour, for which he shared a Nobel prize in 1937 (with Davisson) "for
their experimental discovery of the diffraction of electrons by crystals".
Tipler (5th edition) p 834-839 contains some useful information about the motion of
electrons in uniform magnetic fields.
Thomson's original 1897 paper is reproduced in Classical Scientific Papers S Wright
(Mills and Boon, 1964).
For your extended report
A few points to ponder. What were the limiting errors in this experiment.? What
precaution did you take to minimise random errors? What are the most likely systematic
errors in this experiment. The accepted value for (e/m) is known to 8 significant figures1 .
Would it be possible to repeat such precision with the apparatus used here?
1
See, for example, the National Institute of Standards and Technology web site at http://physics.nist.gov