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Experimental Verification of the Kinematic Equations of Special Relativity and the Mass and Charge of the Electron

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Experimental Verification of the Kinematic Equations of Special Relativity and the Mass and Charge of the Electron

  1. 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 ∗ 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. 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. 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. 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. 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.