The document experimentally verifies the kinematic equations of special relativity and determines the mass and charge of the electron. It describes an experiment that measures the momentum and kinetic energy of electrons over a range of speeds. The data is fitted to both the Newtonian and relativistic kinematic models. The relativistic model provides a much better fit and allows determining the electron charge to mass ratio and mass. The values found agree well with accepted values, supporting the validity of special relativity.

1. Experimental Verification of the Kinematic Equations of Special Relativity and the
Mass and Charge of the Electron
Daniel A. Bulhosa∗
MIT Department of Physics
(Dated: December 4, 2013)
We demostrate the failure of Newtonian kinematics to describe the relationships between kinetic
energy, momentum, and speed for high speed particles. We also demostrate the success of special
relativity to make the correct predictions in this regime. In the process of fitting the relativistic
models to the data we determine the electron charge to mass ratio and the electron mass, which
allow us to determine the elementary charge. The values we find agree fairly well with the accepted
ones.
I. INTRODUCTION
In the mid-19th Century James Clark Maxwell unified
the contemporary knowledge about electricity and mag-
netism by condensing it into the four famous equations
bearing his namesake. The vacuum solution to these
equations predicted the propagation of transverse electric
and magnetic fields through space at a rate determined
by fixed fundamental constants of nature [1]. Maxwell’s
equations did not predict any variation of the speed of
light across the frames of observers moving relative to one
another, which disagreed with the well-established laws
of kinematics posited by Newton in the 17th Century.
Attempts were made by theoretical physicists in the
late 19th Century to explain this discrepancy, including
the proposition that Maxwell’s equations only applied in
the frame of some medium whose disturbance was the
source of electromagnetic phenomena. The existence of
this medium, known as the luminiferous aether, was first
empirically challenged by Albert Michelson and Edward
Morley in 1887, who carried out an experiment which
in theory was capable of detecting motion relative to the
stationary aether. The failure of Michelson and Morley to
detect the aether led to further research that culminated
with Einstein’s proposal of the Special Theory of Rela-
tivity. Through the simple postulation that the laws of
physics and the speed of light are the same in any frame,
Einstein discarded the notion of the aether and general-
ized the kinematic equations of Newton to the realm of
fast speed in one fell swoop [2].
In these experiments we test the kinematic equations
implied by Einstein’s postulates, and compare their pre-
dictions to those of pre-relativity models. In the course of
these tests we derive the mass and the charge of the elec-
tron e, both of which are some of the most fundamental
constants of nature.
II. THEORY
Newton’s laws of motion succesfully predicted the re-
lationships between speed and kinetic energy, and speed
∗ dbulhosa@mit.edu
and momentum within the realm of energies accessible to
physicists before the 19th Century. These relationships
are described by the familiar equations:
p = mv, K =
1
2
mv2
(1)
Here m is the mass of the object in question, and v is its
speed.
Einstein’s postulates predict a correction term γ which
is related to the speed of the particle [3]:
γ =
1
1 − v2
c2
(2)
The generalized equations for the momentum and kinetic
energy of an object of mass m predicted by the postulated
are then:
p = γmv, K = mc2
(γ − 1) (3)
It is clear from equation 2 that γ ≈ 1 when v is small, so
that in this regime the equation for momentum 3 clearly
agrees with equation 1. The equations for kinetic energy
also agree in this regime; this can be confirmed upon
Taylor expanding the relativistic expression.
III. EXPERIMENTAL APPARATUS
III.1. Set-up and Signal Chain
The goal of our experiment was to measure the depen-
dence of momentum and kinetic energy on speed, and to
test the predictions of the above sets of equations. For
this purpose we made use of the apparatus shown in Fig-
ure 1.
The apparatus consisted of a spherical arragement
of wire containing an evacuated cylindrical chamber.
Within the chamber lay a radioactive sample of Stron-
tium 90, which served as a source of electrons through
beta decay. Diametrically opposite to the source was a
set of long charged plates, and a PIN diode that served

2. 2
FIG. 1. Figure adapted from [4]. The Strontium 90 source is
shown as a black box at the bottom, and the velocity selector
and diode detector are labelled by VS and PIN respectively.
Adjustable voltage and current supplies were used to set the
electric and magnetic field values respectively.
as an electron detector. When a constant current was
passed through the wire a uniform magnetic field was
generated in the plane of the source and the detector.
Electrons traveling in this uniform magnetic field expe-
rienced a centripetal force, whose describing equation [1]
implies the following relationship between the strength
of the magnetic field B, the momentum of the electron,
and the radius r of the circle along which it travels:
p = erB (4)
Between the electron source and the detector there
was a collimator that only let through electrons trav-
elling along a circle of radius approximately 20.3 ± 0.2
cm. Given the applied magnetic field equation 4 implies
that the collimator only allowed through electrons with a
narrow band of momenta centered at the value predicted
by the equation. The electrons with the selected band
of momenta continued to travel in a circle until they ar-
rived at the set of long charged parallel plates, which had
a uniform electric field E = V/d between them, where
d is the plate separation and V is the applied voltage.
In order to arrive at the detector the electrons had to
travel approximately in a straight path after reaching the
plates, otherwise they would collide and be collected by
the plates. This occured when the electrons experience
no net force while traveling through the plates, which the
Lorentz force law predicts to be the case when:
v =
E
B
(5)
Electrons with approximately the appropiate speed ar-
rived to the detector, where excitations proportional to
their respective kinetic energies were created and sent to
the preamplifier and amplifier. The preamplifier and am-
plifier converted the excitation into voltage pulses whose
height was proportonial to the kinetic energy of the elec-
trons, and the heights and numbers of these pulses were
recorded by a multi-channel analyzer.
For a chosen magnetic field (and thus chosen momen-
tum), we determined the electric field value (and thus
velocity) which maximized the rate of arrival of electrons
to the detector. In this way, through equations 4 and 5,
the apparatus allowed us to measure the kinematic re-
lationship between speed and momentum. Similarly, by
taking the mean of the distribution of energies measured
for some choice of electric and magnetic field values we
were able to measure the relationship between velocity
and kinetic energy.
III.2. Calibration
FIG. 2. The spectrum of Barium 133 as recorded by the PIN
detector and MCA over the course of three days. We fitted
a line through the three visible features of the spectrum (at
bins 175.34, 454.54, and 1699.62), yielding the correspondence
Energy (keV) = 0.1783 (keV/bin) · (Bin#) − 0.1949 keV.
The multi-chanel analyzer (MCA) records any voltage
pulse with amplitude between 0 and 10 V and records it
in one of 2048 equally sized bins. Pulses with amplitudes
outside of this range are not recorded. Thus for optimal
precision it is best to choose amplifier settings such that
the limits of the range of kinetic energies one expects to
observe correspond to voltage pulse amplitudes of about
0 and 10 V. In order to determine these settings we placed
a Barium 133 source with a known spectrum [5] near
the detector and let the detector and MCA collect and
record this spectrum over the course of about 10 minutes.
We determined a coarse gain of 3000x and a fine gain
of 0.7x to be good settings for this purpose. We also
choose the smallest pulse width setting (0.5 µs) so that
pulses due to different electrons would not overlap and
be undercounted by the MCA.
Having fixed the amplifier settings we ran a long cal-
ibration run (66 hours) so that the peaks of the known
spectrum would be very well defined. Fitting through
the values of known features of the spectrum allowed us
to derive a linear regression determining the correspon-
dence between different kinetic energy values and MCA
bin numbers. We used this regression to calibrate the
MCA. Due to the uncertainty in the location of the peaks
there was a systematic error of ±2 keV associated with

3. 3
our measured values of energy. A plot of the spectrum
recorded through the long calibration is shown in Figure
2.
The calibration of the gaussmeter that we used for the
experiment proved to be unreliable over long periods of
time. The relationship between the applied current and
the resulting magnetic field however is strongly linear,
so rather than measuring magnetic field values over the
long course of the experiment with the gaussmeter we
empirically determined the relation between applied cur-
rent and magnetic field over a short period of time. We
then fit this relation with a line and used the linear fit
to convert the input current values to accurate magnetic
field values.
IV. PROCEDURE AND DATA PRESENTATION
For eight different magnetic field values we measured
the spectrum of kinetic energies and count rates corre-
sponding to multiple choices for the electric field value
(about seven on average) centered about the field value
predicted by the relativistic model. The raw data
recorded by the MCA is shown in Figure 3. For each
choice of magnetic and electric field we let the MCA col-
lect data for 3 minutes. The error bars were determined
by assuming that number of counts corresponding to a
particular energy value is a Poisson random variable.
FIG. 3. Distribution of electron energies counted by the de-
tector and recorded by the MCA for a particular choice of
electric and magnetic field values. The peak of this distribu-
tion, as determined by fitting, is located at 293 keV. For the
different field value settings the discriminator was set between
bins 95 and 110.
After each run the MCA displayed the total number
of electrons it counted for its duration. By dividing this
number over the running time we were able to determine
the counting rate corresponding to given values of the
fields. Figure 4 shows a plot of the count rates corre-
sponding to different voltage values for B = 57.72 G.
As we expect, there is a maximum count rate, which oc-
curs when we select the voltage (and thus velocity) corre-
sponding to the appropiate momentum. For a given mag-
netic field in order to take into account statistical error,
we took the voltage Vmax corresponding to the maximum
count rate to be the mean of the voltages correspond-
ing to count rates whose value lay within the error bar
(1σ) of the highest measured count rate value (see Fig-
ure 4). There were always two data points immediately
outside the group of points used to determine Vmax by
the above criterion. We took the largest of the distances
between Vmax and the voltages of corresponding to these
two points to be the error in Vmax. A table containing
the Vmax values we determined for each B can be found
in the appendix.
FIG. 4. A plot of the count rate as a function of the voltage
applied to the plates. The maximum count rate data point
was at 1.8 kV, so we used the points above the green bar to
calculate Vmax = 1.73. By the criterion explained we took the
error to be the distance between Vmax and the voltage value
of the data point on the left (located at 1.5 kV).
Combining the relativistic equation 3 for p(v) with
equations 4 and 5 yields the following prediction for the
relationship between E and E/B where E = V/d:
B =
m
er
Vmax(B)
Bd
1 −
Vmax(B)
cBd
−1/2
(6)
On the other hand the classical equation 1 for p(v) pre-
dicts the same relation as equation 6 without the term in
square brackets, which is in fact equal to the relativistic
correction term γ.
Having found Vmax for each of the eight magnetic field
values, we fitted both of these models to our data sets,
with m/e as our fitting parameter. The data and the fits
for both models is shown in Figure 5. The vertical error
bars all have length 1.21 G, where 1.00 G was contributed
by the uncertainty in the magnetic field value of the lab
calibration source (1% of 99.96 G), and 0.21 G was con-
tributed by the uncertainty in the current through the
linear fit. We can see that the relativistic model fits the
data significantly better than the classical one.
The value of the charge to mass of the electron pre-
dicted by the relativistic model is e/m = (1.73 ± 0.01) ·
1011
C/kg. The accepted value for e/m is 1.76·1011
C/kg
[6], which is three standard deviations away from our
value so we have fairly good agreement. The classical
model predicts e/m to be (1.19 ± 0.01) · 1011
C/kg, for
which the accepted value is over sixty standard deviations
away! The success of the relativistic theory in fitting the
data and predicting the correct value for this fundamen-
tal ratio provides strong support for its validity.

4. 4
FIG. 5. A plot of our measured data and our best classical
and relativistic fits. As we can see the relativistic model fits
the data relatively well whereas the classical models disagrees
drastically. We used the Junior Lab fitting code, which is
based on an algorithm from [7]. The code returns the χ2
of
the fit, as well as the best fit parameters and their errors.
In order to determine the relationship between velocity
and kinetic energy we fit a gaussian distribution through
the kinetic energy distribution corresponding each choice
of field values. The choice to fit a gaussian is reasonable
if it is assumed that the band of electron velocities that
make it through the plates and into the detector is small
compared to the error induced in the measured velocity
due to thermal noise in the detector. We made this as-
sumption, and for each magnetic field value B we took
the kinetic energy to be equal to the average of the means
of the gaussians fitted to the distributions we measured
for different voltage values. We took the error in the ki-
netic energy to be the average of the standard deviations
of the same gaussians. We chose to take this mean over
the different voltage values, since we observed no relation
between voltage and kinetic energy for a given value of B
that suggested we should favor a particular subset of the
voltages. A table summarizing the kinetic energy values
thus determined can be found in the appendix.
Figure 6 shows the resulting plot of the kinetic energy
as a function of γ − 1, where by equations 2 and 5:
γ = 1 −
Vmax(B)
cBd
2 −1/2
(7)
Here we used Vmax(B) as our voltage value, since this
is the electric field which selects velocities corresponding
to the momenta already selected by B. Figure 7 shows
a plot of the same data, but this time in terms of β2
against the kinetic energy. The two figures are fit by the
relativistic and classical models respectively. We can see
that the relativistic model fits significantly better again.
The relativistic model predicts the electron mass to
be m = 525.9 ± 3.8 keV/c2
. The accepted value of 511
keV [6] is four standard deviations from our measured
value so we have relatively good agreement. The classical
model conversely predicts an electron mass of 914.0 ±
6.6 keV/c2
, for which the accepted value lies over sixty
FIG. 6. A plot of the kinetic energy versus speed data ex-
pressed in terms of γ − 1. The line fit is the relativistic model
for this kinematic relationship as described by equation 8.
The χ2
/dof for this fit was 8.10.
FIG. 7. A plot of the kinetic energy versus speed data ex-
pressed in terms of β2
. The line fit is the classical model for
this kinematic relationship as described by equation 1 in the
form K = 1
2
mc2
β2
. The χ2
/dof for this fit was 25.
standard deviations away. The relativistic model again
significantly outperforms the classical one.
We can multiply the values we determined for the elec-
tron charge to mass ratio and the electron mass with the
relativistic models to determine the electron charge and
the associated propaged error. Our experiment predicts
that e = (1.62±0.01)·10−19
C whereas the accepted value
is e = (1.60 ± 0.01) · 10−19
C [6], which agree within an
error of two standard deviations.
V. CONCLUSIONS
We carried out experiments to test the accuracy of clas-
sical and relativistic models in describing the kinematics
of high speed electrons. The classical models performed
very poorly at fitting our measured data, whereas the rel-
ativistic models performed relativitely well and allowed
us to determine the values of the mass and charge of the
electron fairly accurately (within a few standard devia-
tions). The success of relativity in our experiments reaf-
firms its place as the more general theory of kinematics.

5. 5
[1] E. M. Purcell, Electricity and Magnetism (McGraw-Hill
Book Company, 1985).
[2] W. Rindler, Relativity: Special, General, and Cosmologi-
cal (Oxford University Press, Inc., 2006).
[3] A. P. French, Special Relativity (MIT Press, 1968).
[4] Relativistic Dynamics: The Relations Among Energy, Mo-
mentum, and Velocity of Electrons and the Measurement
of e/m, MIT Department of Physics (2013).
[5] R. Firestone, “Exploring the table of isotopes,” Accessed
on Nov. 18th, 2013.
[6] Wolfram Research. “Elementary charge, electron mass,
electron charge to mass ratio,” Accessed on Dec. 3rd,
2013.
[7] P. Bevington and D. Robinson, Data Reduction and Error
Analysis for the Physical Sciences (McGraw-Hill, 2003).
VI. APPENDIX: DATA SUMMARIES
I(A) B(G) Vmax(kV) σV (kV)
2.74 57.72 1.73 0.23
3.25 68.39 2.25 0.25
3.70 77.81 2.85 0.15
3.93 82.62 3.20 0.10
4.15 87.22 3.35 0.35
4.40 92.45 3.68 0.18
4.62 97.05 3.95 0.15
4.87 102.29 4.15 0.15
TABLE I. A summary of the magnetic field values and the
corresponding voltage values that yielded the maximum count
rate. The uncertainties in the voltages are also shown.
B (Gauss) Mean[µKE] (keV) Mean[σKE] (keV)
57.72 77.57 47.91
77.81 186.30 20.27
82.62 206.84 4.34
87.22 226.41 4.44
92.45 250.07 4.38
97.05 272.09 4.51
102.29 293.77 4.16
TABLE II. A summary of the kinetic energies corresponding
to the inputed magnetic field values. These were determined
by taking averaging over the means of the distributions cor-
responding to one value of B.