Determination of the Specific Charge of the Electron
1. Purpose





Study Cathode rays.
Study Lorentz force.
Study of the deflection of electrons in a magnetic field into a circular orbit.
Determination of the magnetic field B as a function of the acceleration potential U of the
electrons at a constant radius r.
Determination of the specific charge of the electron.
2. Historical and theoretical background
J. J. Thomson first determined the specific charge e/me of the electron in 1897.
In his experiment (Fig. 1), J.J. Thomson had found a charged particle that had a
specific charge two thousand times greater than that of the hydrogen ion, the
lightest particle known in 1897. Once the charge on the particles was measured
Fig 1
he could say with certainty that they were two thousand times lighter than
hydrogen. This explained how they could pass through thin sheets of gold. Particles this small
could pass between atoms in a solid.
J.J. used aluminum to make most of his electrodes. He repeated his experiments using cathodes
made from different metals, including iron and platinum, and found that the specific charge did not
change. He argued that the cathode ray consisted of small charged particles, and by using different
types of cathodes realized that the particles existed in many types of atoms. He concluded that the
particles, which he called 'corpuscles', were a universal constituent of matter - they form part of all
the atoms in the universe. We now know these particles as electrons.
For his work he received The Nobel Prize in Physics in 1906 “in recognition of the great merits of
his theoretical and experimental investigations on the conduction of electricity by gases”.
(http://nobelprize.org/nobel_prizes/physics/laureates/1906/thomson-bio.html )
The mass me of the electron is hard to come by experimentally.
It is easier to determine the specific charge of the electron e/m
from which the mass m can be calculated if the elementary
charge e is known:
An electron moving at velocity v perpendicularly to a
homogenous magnetic field B, is subject to the Lorentz force F:

 
F  e  (v xB) (1)
which is perpendicular to the velocity and to the magnetic field.
As a centripetal force
v2
F  me 
(2)
r
it forces the electron into an orbit of radius r (see Fig. 2), thus
Fig. 2
e
v

(3)
me r  B
In the experiment, the electrons are accelerated in a fine beam tube by the potential U. The
me  v 2
resulting kinetic energy is e  U 
(4) The specific charge of the electron thus is
2
1
e
2 U

(5) The magnetic field B is generated in a pair of Helmholtz coils and is
me ( r  B ) 2
proportional to the current I in the Helmholtz coils: B  k  I (6)
e
2 U

(7)
me ( r  k  I ) 2
The dependence on the accelerating potential U of the current I, in the magnetic field of which the
orbiting radius of the electrons is kept to a constant value r, follows after recasting equations (VI)
3
4
n
e 1 2 2 2
  r  k  I (8) The proportionality factor is k   0  ( ) 2  (9)
and U 
where
5
R
me 2
 0  4    10 7 Vs / Am (10) can be calculated either from the coil radius R = 150 mm and the
winding factor n = 130 per coil, or be determined by recording a calibration curve B = f(I). All
determining factors for the specific electron charge are now known.
3. Setup
3.1. Introduction
Electrons are accelerated in an electric
field and enter a magnetic field at right
angles to the direction of motion. The
specific charge of the electron is
determined from the accelerating
voltage, the magnetic field strength
and the radius of the electron orbit.
The fine beam tube contains hydrogen
molecules at low pressure, which
through collisions with electrons are
caused to emit light. This makes the
orbit of the electrons indirectly visible,
and their orbiting radius r can be
directly measured with a ruler.
Fig. 3
3.2. Description of Equipment
Technical data
1 Fine beam tube (Fig. 3)
1 Helmholtz coils with holder and measuring
device
1 DC power supply 0 … 500 V
1 DC power supply 0 … 20 V
1 Voltmeter, DC, <= 300 V
1 Ammeter, DC, <= 3 A
1 Steel tape measure, 2 m
3 Safety connecting leads, 25 cm
3 Safety connecting leads, 50 cm
7 Safety connecting leads, 100 cm
Fig. 4
additionally recommended:
1 Teslameter, 1 Axial B-probe (Fig. 4), 1 Multicore cable, 6-pole, 1,5 m long
2
Fine beam tube specifications:
Gas filling: hydrogen, approx. 1 Pa
Electron beam system: indirectly heated oxide
cathode, Wehnelt cylinder, conical anode with
semi-cylindrical screen
Heating voltage and current: 6.3 V, approx. 1 A
Anode voltage U :150 V DC to 300 V DC
Wehnelt voltage UW :max. 10 V
Pair of plates for electrostatic deflection
(directly behind anode)
Plate voltage UP :50 V DC to 100 V DC
Helmholtz coils
Number of turns n: 130 per coil
maximum permissible coil current IS :2 A
Fig. 5
(briefly 3 A)
Resistance R:approx. 2 Ω per coil
Coil radius r:150 mm
Coil spacing: a:150 mm
Legend (Fig. 6):
1. Fine beam tube
2. Holder for supporting the fine beam tube and the
coils in a defined position
3. Pair of Helmholtz coils
4. Measuring device ), consisting of support with
two slides and support with mirror, for determining
the diameter of the circular electron beam.
5. 2 clamps, 4 screws, 4 washers
Socket connected with
7 Helmholtz coils
8 Deflection plates
9 Anode
10 Wehnelt cylinder
11 Heating
12 Cathode
Fig. 6
13 Anode
Operating voltages connected to safety sockets 7
to 13, which are connected internally to sockets
E (coil connections) and to the tube via a
permanently attached lead 6 with 6-pin plug
3.3. Assembling the experiment
The experimental setup to determine the specific
electron charge is shown in Fig. 3, the electric
connections in Fig. 7 and Fig. 8.
– Disconnect power supply and turn all rotary
potentiometers to left catch position.
– Connect the 6.3-V input end of the fine beam
tube to the 6.3-V outlet of the DC power supply.
Fig. 7
3
– Short-circuit the positive pole of the 50-V outlet of the DC power supply with the negative pole
of the 500-V outlet and connect with the
socket “-” of the fine beam tube (cathode).
– Connect the socket “+” of the fine beam
tube (anode) with the positive pole of the
500-V outlet, the socket W (Wehneltcylinder) with the negative pole of the 50V outlet.
– In order to measure the acceleration
potential U connect the voltmeter
(measuring range 300 V–) to the 500-V
Fig. 8
outlet.
– Short the deflection plates of the fine beam tube to the anode.
– Connect the DC power supply and ammeter (measuring range 3 A–) in series with the Helmholtz
coils.
– Power up the DC power supply and set acceleration potential U = 300 V.
Thermionic emission starts after warming up for a few minutes.
– Optimize focusing of the electron beam by varying the voltage at the Wehnelt-cylinder from 0 …
10 V until it leads to a narrow, well defined beam with clear edge definition.
– Connect the DC power supply of the Helmholtz coils and look for current I, at which the electron
beam is deflected into a closed orbit.
If the electron beams after leaving the anode is deflected to the wrong (left) side:
– disconnect both power supplies.
– exchange the connections at the DC power supply in order to change the polarization of the
magnetic field.
If the electrons do not move on a closed orbit but on a helical curve line:
– Loosen the mounting bolts of both holding brackets (read the information manual for the fine
beam tube).
– Carefully rotate the fine beam tube around its longitudinal axis, until the electron beam runs on a
closed circular orbit.
– Fasten mounting bolts.
4. Safety precautions
Attention: The fine beam tube requires dangerous contact voltages up to 300 V for accelerating
the electrons. Other voltages that are connected with this dangerous contact voltage also present a
contact hazard. Dangerous contact voltages are thus present at the connection panel of the holder
and at the Helmholtz coils when the fine beam tube is in operation.
 Connect the connection panel only via safety connecting leads.
 Always be sure to switch off all power supplies before connecting and altering the
experiment setup.
 Do not switch on the power supplies until you have finished assembling the circuit.
 Do not touch the experiment setup, particularly the Helmholtz coils, during operation.
Danger of implosions: The fine beam tube is a evacuated glass vessel with thin walls.
to left catch position.
 Do not subject the fine beam tube to mechanical stresses.
 Operate the fine beam tube only in the holder (555 581).
 Connect the 6-pole plug of the holder carefully to the glass base.
 Read the instruction sheet supplied with the fine beam
4
5. Experimental procedures
1. Move the left slide of the measuring device so that
its inner edge, mirror image and escape aperture of
the electron beam come to lay on one line of sight.
2. Set the right slide for both inside edges to have a
distance of 8 cm. (Fig. 9)
3. Sight the inside edge of the right slide, align it with
its mirror image and adjust the coil current I until the
electron beam runs tangentially along the slide edge
covering the mirror
Fig. 9
image (see Fig. 9).
4. Reduce the acceleration potential U in steps of 10 V to 200 V and choose the coil current I so
that the orbit of the electron beam has a diameter of 8 cm.
5. Record acceleration potential U and coil current I
6. Plot on a graph U=f(I2). From the plot determine the slope (α).
e
2 
 2 2
7. Calculate the specific charge of the electron using formula:
me r  k
The proportionality factor k can be calculated:
- either from the coil radius R = 150 mm and the winding factor n = 130 per coil, using formula:
3
4
n
k   0  ( ) 2  where  0  4    10 7 Vs / Am
5
R
- or be determined by recording a calibration curve B = f(I).
Calibration of the Helmholtz magnetic field (optional):
The setup for calibrating the magnetic field
is shown in Fig. 10.
The additionally recommended devices
mentioned above are required for making
measurements.
1. If applicable disconnect all power supply
units.
2. Remove the measuring device and the
Helmholtz coil at the front side, loosen the
connection to the fine beam tube and the
mounting bolts of the two holding brackets
(read the
instructions for the fine beam tube).
3. Carefully remove the fine beam tube and
place it e.g. in its original case.
4. Re-assemble the Helmholtz at the front
side coil and connect.
Fig. 10
5. Connect the axial B-probe to the
Teslameter (measuring range 20 mT) and calibrate the zero-point (see Instruction Manual for
Teslameter).
6. Move the axial B-probe parallel to the magnetic field of the Helmholtz coils into the center of
the pair of coils.
7. Raise the coil current I from 0 to 3 A in steps of 0.5 A, measure the magnetic field B, and record
the measured values.
5
8. After conclusion of the calibration: Re-assemble the fine beam tube according to the
instructions.
Measuring example:
Table 1
U
I
290.1
280.7
269.8
260.1
250
240
230
220
210
200
I*I
1.67
1.66
1.65
1.64
1.62
1.6
1.57
1.52
1.5
1.48
2.7889
2.7556
2.7225
2.6896
2.6244
2.56
2.4649
2.3104
2.25
2.1904
Average
U/I*I
Graph
slope
The coil current I (A) as a function of the accelerating
U(V) potential, the square of the current and the
fraction U/I*I are presented in the table 1.
The data set from the table are made for a ratio r=4
cm of the electron orbit.
The graph in the Fig. 11 presents the measuring result
from Table 1.
The theoretic accepted value for the electronic charge
e
A s
 1.76  10 11
of the electron is
me
kg
U/I*I
104.0195
101.8653
99.10009
96.70583
95.25987
93.75
93.31007
95.22161
93.33333
91.30752
( e  1.6  10 19 A  s ; me  9.1  10 31 kg )
96.38731
Using the value from the table 1 and the formula 7
e
A s
e
A s
96.906
 1.98  10 11
 1.99  10 11
the
and
me
kg
me
kg
The value for k was calculated according to formula (9) and (10).
350
300
y = 96.906x
U (V)
250
200
150
100
50
0
0
0.5
1
1.5
2
2.5
3
I*I (A*A)
Fig. 11
6. Final considerations
If the velocity v of electrons is not
perpendicularly to a homogenous magnetic
field B, the trajectory of the electrons will be a
helix.
Further application of the moving of the
charge in a magnetic field:
 Mass spectrometer: The mass
spectrometer is an instrument which
can measure the masses and relative
Electronic gun
Helix
Fig. 12
6


concentrations of atoms and molecules,
Cyclotron: The cyclotron was one of the earliest types of particle accelerators, and is still
used as the first stage of some large multi-stage particle accelerators. It makes use of the
magnetic force on a moving charge to bend moving charges into a semicircular path
between accelerations by an applied electric field. The applied electric field accelerates
electrons between the "dees" of the magnetic field region. The field is reversed at the
cyclotron frequency to accelerate the electrons back across the gap.
Magnetic confinement: a charged particle will be bent into a circular path by the
component of magnetic field which is perpendicular to the velocity. This confines the
charge in a orbit in a localized region of space so that it doesn't interact with container
walls.
7