Determination of the Specific Charge of the Electron
? 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/m of the electron in 1897. eIn 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”.
The mass m of the electron is hard to come by experimentally. e
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?(vxB) (1)
which is perpendicular to the velocity and to the magnetic field.
As a centripetal force
it forces the electron into an orbit of radius r (see Fig. 2), thus Fig. 2
ev (3) ?mr?Be
In the experiment, the electrons are accelerated in a fine beam tube by the potential U. The
2m?vee?U?resulting kinetic energy is (4) The specific charge of the electron thus is 2
e2?U(5) The magnetic field B is generated in a pair of Helmholtz coils and is ?2m(r?B)e
proportional to the current I in the Helmholtz coils: (6) B?k?I
e2?U (7) ?2m(r?k?I)e
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)
34ne12222?()?????and (8) The proportionality factor is (9) where k???UrkI05Rm2e
?7??4???10Vs/Am (10) can be calculated either from the coil radius R = 150 mm and the 0
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.
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.
3.2. Description of Equipment
1 Fine beam tube (Fig. 3)
1 Helmholtz coils with holder and measuring
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
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
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
10 Wehnelt cylinder
12 Cathode Fig. 6
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. – Short-circuit the positive pole of the 50-V outlet of the DC power supply with the negative Fig. 7
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 (Wehnelt-
cylinder) with the negative pole of the 50-
– In order to measure the acceleration
potential U connect the voltmeter Fig. 8
(measuring range 300 V–) to the 500-V
– 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
– 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
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
– 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
? 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
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).
e2??7. Calculate the specific charge of the electron using formula: ?22mrk?e
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:
34n?72?()??4???10Vs/Am where k???005R- 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
1. If applicable disconnect all power supply
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
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
Fig. 10 side coil and connect.
5. Connect the axial B-probe to the
Teslameter (measuring range 20 mT) and calibrate the zero-point (see Instruction Manual for
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.
8. After conclusion of the calibration: Re-assemble the fine beam tube according to the instructions.
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. U I I*I U/I*I
The data set from the table are made for a ratio r=4 290.1 1.67 2.7889 104.0195
cm of the electron orbit. 280.7 1.66 2.7556 101.8653
The graph in the Fig. 11 presents the measuring result 269.8 1.65 2.7225 99.10009
from Table 1. 260.1 1.64 2.6896 96.70583
250 1.62 2.6244 95.25987 The theoretic accepted value for the electronic charge
240 1.6 2.56 93.75 ?eAs?11??1.7610 of the electron is 230 1.57 2.4649 93.31007 mkge220 1.52 2.3104 95.22161 ??1931m?9.1?10kg(; ) e?1.6?10A?s210 1.5 2.25 93.33333 e
200 1.48 2.1904 91.30752 Using the value from the table 1 and the formula 7
Average ??eAseAs?11?11????1.98101.9910U/I*I 96.38731 the and mkgmkgGraph ee
slope 96.906 The value for k was calculated according to formula (9) and (10).
300y = 96.906x
Fig. 11Fig. 11
Electronic gun 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
Further application of the moving of the
charge in a magnetic field:
? Mass spectrometer: The mass
spectrometer is an instrument which
Helix can measure the masses and relative
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.