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Charge to Mass Ratio of the Electron
1. Purpose:
To determine the charge to mass ratio of the electron, e/m, by subjecting a beam
of electrons to a magnetic field and examining their trajectories. It can also
provide a rough estimate of the magnitude of the earth’s magnetic field.
2. Introduction:
The charge to mass ratio of an electron is measured [1] from observing the
trajectories of electrons in a magnetic field. When an electron moves in an
electrostatic field from point 1 to point 2 with electric potential difference V
between them, the electron will gain a kinetic energy (K) given by
K = mu2/2 = eV.
(1)
where e = charge of the electron, m = mass of the electron,
u = velocity of the electron.
When a charged particle of charge q with velocity u moves through a magnetic
field with magnetic field strength B, it experiences a force ("Lorentz force")
given by
F=quxB
(2)
If B is perpendicular to u then the electron will move in a circle whose plane is
also perpendicular to B, and the magnetic force (Lorentz force) provides the
centripetal force for this circular motion, i.e.
mu2/r = euB
(3)
where m = mass of electron
r = radius of circular path.
Solve Eq. (3) for u and substitute into Eq. (1) to obtain an expression for the
charge-to-mass ratio of the electron
e/m = 2V / (Br)2
(4)
So if we know the magnitude of the magnetic field, the potential difference V and
the radius of the path r, we can calculate the charge-to-mass ratio of the electron.
For electrons (or any given kind of particle) the left hand side is a constant. For
example, if we raise the energy of the electrons injected into the magnetic field
region (by increasing the accelerating potential V) and we desire to maintain an
orbit with the same radius, then this equation informs us we would have to
increase the magnitude of the magnetic field. Alternatively, if we want to increase
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the radius of the circular orbit we would have to decrease the magnitude of the
magnetic field. The experimental problem is to generate a uniform magnetic field.
The experiment involves injecting electrons into a magnetic field generated by
Helmholtz coils, which provide an approximately uniform magnetic field close to
the center, but there is some non-uniformity which may affect the measurement.
3. Apparatus:
3.1 items:
3B ELWE electron beam tube [1],
Helmholtz coils (Leybold),
power supplies,
multimeters.
3.2 Set of coils:
The magnetic field is supplied by the Helmholtz coils. They are constructed so
that the distance between the two loops is approximately equal to their radii
(about 15cm in our case, a=147.5 mm). This arrangement gives maximum
uniformity of the field over a large region near the center. Each of the coils
contains 124 turns. The magnetic field strength at the center is given by (derive
this in your report):
8 
 0
(5)
a 125
where N = number of turns in each coil,
I = current through the coils in amperes,
a = radius of coils in meters, and
o = permeability of free space
The set of coils that we use has a slight deviation from the ideal Helmholtz
configuration: the distance b between the two coils is not exactly equal to the
radius a. As a consequence the relation (5) above giving the magnetic field
strength B0 at the center needs to be modified as follows:
3
2
80  
b  2
Bo  a 4   a  


We may rewrite this in the form
Bo = KI (7)
(6)
With K given by
3
2
80  
b  2
K
4    
a 
 a  
(8)
where K is a constant for a given pair of coils and Bo is in units of Tesla.
2
Because the radius is not equal to the separation, the coils are not an ideal
Helmholtz pair, and the value of K which we have given is slightly different from
the ideal value that you would get for a real Helmholtz coil. Knowing the
magnitude of the field at the center point between the coils is not enough since
the plane containing the electron’s orbit (a plane parallel to the plane of the coils)
does contain the center point, but the orbit will cover points at some distance
from the center. Given the geometry, the field has radial symmetry and it will be
smaller as we move out from the center point. To estimate the field at a distance
R from the center, you can use the data given in the table 1 below. The
normalized distance from the center of the coil R/a is listed with the normalized
field B/Bo, where Bo is the field at the center.
R/a
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
B/B0
1.00000
0.99996
0.99928
0.99621
0.98728
0.96663
0.92525
0.85121
0.73324
0.56991
0.38007
Table 1: normalized field B/Bo vs normalized distance R/a from the center point
of the set of Helmholtz coils used in this experiment.
3.3 Electron beam tube:
The electron trajectories will be measured in a vacuum tube which contains some
neon gas (at a pressure of 1.3 10-5 bar). Some of the electrons excite the neon
atoms and the faint yellow light emitted upon recombination makes the electron
trajectory visible. The pressure of the gas in the tube is selected so that the
space charge produced by the electrons bunches the beam for this range of
velocities. The electrons are generated by heating a filament. Heating gives
some of the electrons in the filament enough energy to overcome the surface
barrier and escape from the metal. These electrons are then accelerated by a
potential applied between the filament (the cathode) and the grid, and the grid
and the anode. The tube contains an axial scale shaped like a ladder, whose
metal “rungs” are oriented such that the circular beam goes through the ladder.
The rungs allow a precise setting of the beam radius when one looks to the
ladder from the lateral side and opposite rungs are aligned in the line of sight.
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4. Procedure:
4.1 Preparations:
First measure the radius and spacing of the Helmholtz coils, and estimate the
precision with which you determine them.
Make the connections as indicated in figure 1:
Fig. 1: Electrical connections to the coil and neon tube. The cathode (blue) is at negative
potential relative (50V max) to the grid (black) while the anode (red) is at positive
potential relative to the grid (max 300V). The filament heating voltage (green) should be
around 8 V. The coil leads are both shown in black.
Fig.2 Schematics of the connections showing that the electrons are accelerated from the
cathode to the anode. The total potential drop is the voltage V of Eqs 1 and 4.
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Before turning on the power supply, make sure that the two voltages for the
cathode and the anode are both turned down to zero. This avoids voltage being
present at the cathode or anode when the filament voltage is switched on.
Otherwise the cathode layer could be damaged during heating. Wait for at least
a minute of heating time before turning up the voltages to cathode and
anode. Choose -40V for the cathode and +100V for the anode as starting values.
Note that cathode voltage should not exceed -50V, and anode voltage must not
exceed +300V! The full intensity of the electron beam is generally not achieved
until heating has been continued for 2 to 3 minutes.
4.2 Measurement procedure:
In order to account for the Earth’s magnetic field [2], we shall do the
measurements with the setup in two orientations, rotated by 180 o (around a
vertical axis) with respect to each other.
(1) Set the grid-to-cathode voltage to 10V (adjust it up to max 50V to get a clear
and narrow beam) and the grid-to-anode voltage to 200 V (slowly); increase
the current in the Helmholtz coil until the radius of the electron orbit (r) is 5
cm. Turn the tube gently in its clamps about its vertical axis so that the beam
forms a closed circle rather than a spiral.
(2) Increase the current in the Helmholtz coil so as to change the radius of the
orbit to 4,3,..cm., but do not exceed the maximum allowed current in the
Helmholtz coil (3A short-term, 2A long-term)
(3) Increase the anode voltage to 300 V in 20 V increments, for every anode
voltage repeating steps (1) and (2) for r = 5, 4, 3 cm.
(4) Turn down anode and cathode voltages and carefully turn the whole setup
(coils + tube) by 180 degrees (this may require disconnecting some cables)
(5) Repeat steps (1) to (3) with this reversed orientation.
(6) When you flip the orientation of the tube you must still apply the same total B
field (Btotal) in both cases, if the radius of the orbit is the same. However, you
will find that the currents you apply to maintain the electrons in orbits of the
same radius are different. Let us call the larger of the two currents Ibig and the
smaller one Ismall. Obviously, in the first case the earth’s magnetic field Bearth is
opposing the field generated by the coil and in the second case the earth’s
field is adding to the field generated by the coil. Considering both cases in
turn, we have
Btot = Bbig − Bearth = KIbig - Bearth (9),
and
Btot = Bsmall + Bearth = KIsmall + Bearth (10).
These can be added to give
Btot = (K/2) (Ismall + Ibig), (11)
and subtracted to give
Bearth = (K/2) (Ibig - Ismall) (12)
Use Btot to calculate e/m for each measurement pair.
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After the experiment, do not disconnect the leads, but ensure that the power
supply and the coil current are switched off.
5. Analysis:
5.1 Every set of magnet current Itot (and thus magnetic field Btot), accelerating
voltage V and orbit radius r provides an independent measurement of e/m.
(If the two voltages for the two positions (original and rotated) are not
identical, then use the average of the two in the calculation)
5.2 Estimate the uncertainties on your measurements of the directly measured
quantities V, I, r, and a, and from these determine an uncertainty for every
individual measurement of e/m. Take the average of all of these e/m values
and determine the standard deviation. Compare the average of the individual
uncertainties with the standard deviation. (If the uncertainties of the individual
e/m measurements are very different, then it is recommended to use a
weighted average rather than the straight mean (see statistics hand-out).
5.3 You should also plot V (on ordinate) vs. (Br)2 (on abscissa) and determine
e/m from the slope of the straight line. Do this separately for each of the four
data sets corresponding to the four different orbit radii. If you find a non-zero
intercept, try to interpret what this means. Compare the four values -- are
they compatible with each other? If they are different, try to find an
explanation for the difference. Can you correct for the effect which causes
the difference? If not, how should you deal with this when quoting your
result?
Determine the uncertainty on the e/m value obtained from the slopes, and
compare with what you obtained in 5.2
5.4 Compare your results for e/m with the accepted value. Do they agree within
experimental uncertainties?
5.5 Calculate the component of the earth’s magnetic field normal to the plane of
the Helmholtz pair by averaging all estimates of Bearth.. Estimate the
uncertainty using the standard deviation. Note the orientation of the coils with
respect to the Earth’s magnetic field (use the compass). Using the measured
orientation of the coil, estimate the Earth field component that you expect to
find and compare
6. References
[1] 3B Scientific Physics experiment overview posted on the course website
[2] NOAA Earth magnetic field computation:
http://www.ngdc.noaa.gov/geomagmodels/IGRFWMM.jsp
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7. Appendix: Technical specifications
7.1 Helmholtz coil
manufacturer
Model number
Number of coil windings
Coil current
Resistance of coil
Coil radius
Coils spacing
Leybold
555 58 (old version)
124
Max 2A continuous
3A short term
Approx. 5 per coil
Approx. 148 mm
Approx. 151 mm
7.2 Electron beam tube
manufacturer
Model number
Gas filling
Gas pressure
Cathode filament voltage
Cathode voltage
Grid voltage
Anode voltage
Orbit radii corresponding to rungs
Diameter of glass bulb
Total height
7
3B ELWE
U8481430
neon
1.3 10-5 bar
~8 V dc
Max - 50V
0
200 to 300 V
1, 2, 3, 4, 5 cm
160 mm
260 mm