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| [[File:Walter Kaufmann measurement of electron charge to mass ratio.svg|thumb|350px|Figure 1. Walter Kaufmann's measurement of the electron charge to mass ratio for different velocities of the electron. A radium source at the bottom of the evacuated apparatus emitted beta particles of varying energies. Parallel ''E'' and ''B'' fields in combination with the pinhole aperture allowed only certain combinations of electron direction and speed to expose the photographic plate at the top. (a) This front view of the apparatus illustrates the uniform acceleration imposed by the charged condenser plates on the beta particles. (b) This side view of the apparatus illustrates the circular motion of the beta particles transverse to the ''B'' field. (c) The photographic plate recorded a curved streak which was analyzed to determine ''e/m'' for varying ''v'' following different theoretical assumptions.]]
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| The '''Kaufmann–Bucherer–Neumann experiments''' measured the dependence of the [[inertial mass]] (or [[momentum]]) of an object on its [[velocity]]. The [[History of science|historical]] importance of this series of [[experiment]]s performed by various [[physicist]]s between 1901 and 1915 is due to the results being used to test the predictions of [[special relativity]]. The developing precision and data analysis of these experiments and the resulting influence on [[theoretical physics]] during those years is still a topic of active historical discussion, since the early experimental results at first contradicted [[Albert Einstein|Einstein]]'s then newly published theory, but later versions of this experiment confirmed it. For modern experiments of that kind, see [[Tests of relativistic energy and momentum]], for general information see [[Tests of special relativity]].
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| ==Historical context==
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| In 1896, [[Henri Becquerel]] discovered [[radioactive decay]] in a variety of [[chemical element]]s. Subsequently, the [[beta decay|beta radiation]] from these decays was discovered to be the emission of a negatively [[beta particle|charged particle]]. Later these particles were identified with the [[electron]], discovered in [[cathode ray]] experiments by [[J. J. Thomson]] in 1897.
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| This was connected with the theoretical prediction of the [[electromagnetic mass]] by J. J. Thomson in 1881, who showed that the electromagnetic energy contributes to the mass of a moving [[electric charge|charged body]].<ref>{{Citation|author=Thomson, J. J.|year=1881|title=[[s:On the Effects produced by the Motion of Electrified Bodies|On the Effects produced by the Motion of Electrified Bodies]]|journal=Philosophical Magazine|volume=11|series=5|issue=68|pages=229–249}}</ref> Thomson (1893) and [[George Frederick Charles Searle]] (1897) also calculated that this mass depends on velocity, and that it becomes infinitely great when the body moves at the speed of light with respect to the [[luminiferous aether]].<ref name=searle>{{Citation|author=Searle, G.F.C|year=1897|title=[[s:On the Steady Motion of an Electrified Ellipsoid|On the Steady Motion of an Electrified Ellipsoid]]|journal=Philosophical Magazine|series=5|volume=44|issue=269|pages=329–341}}</ref> Also [[Hendrik Antoon Lorentz]] (1899, 1900) assumed such a velocity dependence as a consequence of his theory of electrons.<ref>{{Citation|author=Lorentz, H.A.|year=1900|title=[[s:de:Über die scheinbare Masse der Ionen|Über die scheinbare Masse der Ionen]] ([[s:Translation:On the Apparent Mass of the Ions|On the Apparent Mass of the Ions]])|journal=Physikalische Zeitschrift|volume=2|issue=5|pages=78–80}}</ref> At this time, the electromagnetic mass was separated into "transverse" and "longitudinal" mass, and was sometimes denoted as "apparent mass", while the invariant Newtonian mass was denoted as "real mass".<ref group=A>Miller (1981), pp. 45–47</ref><ref group=A>Pais (1982), pp. 155–159</ref> On the other hand, it was the belief of the German theoretician [[Max Abraham]] that all mass would ultimately prove to be of electromagnetic origin, and that Newtonian mechanics would become subsumed into the laws of electrodynamics.<ref group=A>Miller (1981), pp. 55–67</ref>
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| The concept of (transverse) electromagnetic mass <math>m_T</math>, which was based on specific models of the electron, was later transmuted into the purely kinematical concept of [[relativistic mass]] which concerns all forms of energy, not only electromagnetic energy. Currently, however, the concept of relativistic mass, although still frequently referred to in popular works on relativity, is now seldom used among professional working physicists, and has been superseded by the expressions for [[Mass in special relativity|relativistic energy and momentum]], which also predict that the speed of light cannot be reached by massive bodies. This is because all of those relations involve the [[Lorentz factor]]:
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| :<math>\frac{m_{T}}{m}=\frac{p}{mv}=\frac{E}{mc^{2}}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}</math>
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| Therefore, the Bucherer–Kaufmann–Neumann experiments can be seen as early [[tests of relativistic energy and momentum]]. (For the following historical description of the experiments, the concepts of "transverse" or "relativistic mass" are still used).
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| ==Kaufmann's experiments==
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| ===First experiments===
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| [[File:KaufmannExpGraph1.svg|thumb|200px|Figure 2. Kaufmann's measurements of 1901 (corrected in 1902) showed that the charge-to-mass ratio diminishes and thus the electron's momentum (or mass) increases with velocity. Note that <math>\scriptstyle \epsilon/m_{0}\sim1.76\times10^{7}</math> emu/gm when the electron is at rest.]]
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| [[Walter Kaufmann (physicist)|Walter Kaufmann]] began to experiment with [[beta ray]]s using a device similar to a [[cathode ray tube]], where the source of the electrons was the decay of [[radium]] that was placed in an [[vacuum|evacuated]] container. (See Fig. 1) Such rays emitted from radium were called "Becquerel rays" at that time. Contrary to the then known [[cathode ray]]s which reached speeds only up to 0.3''c, c'' being the speed of light, Becquerel rays reached velocities up to 0.9''c''. However, since beta particles have different velocities, the radiation was inhomogeneous. Therefore, Kaufmann applied [[Electric field|electric]] and [[Magnetism|magnetic]] fields aligned ''parallel'' to each other, so that the deflections caused by them were perpendicular to each other. Their impacts on a photographic plate produced a deflection curve, whose individual points corresponded to a certain velocity and a certain mass of the electrons. By reversing the charge on the condenser, thus inverting the electric field, two symmetric curves could be obtained, whose center line determined the direction of the magnetic deflection.<ref group=A>Miller (1981), pp. 47-54</ref><ref group=A>Staley (2009), pp. 223–233</ref>
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| Kaufmann published a first analysis of his data in 1901 – he actually was able to measure a decrease of the [[Mass-to-charge ratio|charge-to-mass ratio]], thus demonstrating that mass or momentum increases with velocity.<ref>{{Citation | author=Kaufmann, W. | year=1901 | title=[[wikilivres:Die magnetische und elektrische Ablenkbarkeit der Bequerelstrahlen|Die magnetische und elektrische Ablenkbarkeit der Bequerelstrahlen und die scheinbare Masse der Elektronen]] | journal=Göttinger Nachrichten | issue=2 |pages =143–168}}</ref> Using Searle's formula (1897) for the electromagnetic energy increase of charged bodies with velocity, he calculated the increase of the electron's [[electromagnetic mass]] as a function of velocity:
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| :<math>\phi(\beta)=\frac{3}{4\beta^{2}}\left[\frac{1}{\beta}\lg\frac{1-\beta}{1+\beta}+\frac{2}{1-\beta^{2}}\right],\;\beta=\frac{v}{c}</math>,
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| Kaufmann noticed that the observed increase cannot be explained by this formula, so he separated the measured ''total mass'' into a ''mechanical (true) mass'' and an ''electromagnetic (apparent) mass'', the mechanical mass being considerably greater than the electromagnetic one. However, he made two mistakes: As shown by [[Max Abraham]], Kaufmann overlooked that Searle's formula only applies in the longitudinal direction, but for deflection measurements the formula for the transverse direction was important. Therefore Abraham introduced the "transverse electromagnetic mass" with the following velocity dependence:
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| :<math>\phi(\beta)=\frac{3}{4\beta^{2}}\left(\frac{1+\beta^{2}}{2\beta}\lg\frac{1+\beta}{1-\beta}-1\right), </math>
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| Kaufmann also made a calculation mistake in deriving the deflection curves. Those errors were corrected by him in 1902.<ref>{{Citation | author=Kaufmann, W. | year=1902 | title=Über die elektromagnetische Masse des Elektrons | journal=Göttinger Nachrichten | issue=5 |pages =291–296| url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002499444}}</ref>
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| In 1902 and 1903 Kaufmann performed another series of tests with updated and improved experimental techniques. The results were interpreted by him as a confirmation of Abraham's theory and of the assumption that the electron's mass is completely of electromagnetic origin.<ref>{{Citation | author=Kaufmann, W. | year=1902 | title=Die elektromagnetische Masse des Elektrons ([[s:Translation:The Electromagnetic Mass of the Electron|The Electromagnetic Mass of the Electron]]) | journal=Physikalische Zeitschrift | volume =4 | issue=1b |pages =54–56}}</ref><ref>{{Citation | author=Kaufmann, W. | year=1903 | title={{Internet Archive|nachrichten00klasgoog|Über die "Elektromagnetische Masse" der Elektronen}} | journal=Göttinger Nachrichten | issue=3 |pages =90–103}}</ref>
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| [[Hermann Starke]] conducted similar measurements in 1903, although he used cathode rays limited to 0.3c. The results that he obtained were interpreted by him as being in agreement with those of Kaufmann.<ref>{{Cite journal|author=Starke, H.|year=1903|title=Über die elektrische und magnetische Ablenkung schneller Kathodenstrahlen |journal=Verhandlungen der Deutschen Physikalischen Gesellschaft|issue=13|pages=241–250}}</ref>
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| ===Competing theories===
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| [[File:Abraham-Lorentz-Bucherer.svg|thumb|300px|Figure 3. Predictions of speed dependence of transverse electromagnetic mass according to the theories of Abraham, Lorentz, and Bucherer.]]
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| In 1902, [[Max Abraham]] published a theory based on the assumption that the electron was a rigid, perfect [[sphere]], with its charge being distributed evenly on its surface. As explained above, he introduced the so-called "transverse electromagnetic mass" besides the "longitudinal electromagnetic mass", and argued that the entire electron mass is of electromagnetic origin.<ref group=A>Miller (1981), pp. 55–67</ref><ref group=A>Staley (2008), pp. 229–233</ref><ref>{{Cite journal | last=Abraham, M. | year=1902 | title=[[s:de:Dynamik des Electrons|Dynamik des Electrons]]| journal=Göttinger Nachrichten|pages =20–41}}</ref><ref>{{Cite journal | last=Abraham, M. | year=1902 | title=Prinzipien der Dynamik des Elektrons ([[s:Translation:Principles of the Dynamics of the Electron (1902)|Principles of the Dynamics of the Electron (1902)]])| journal=Physikalische Zeitschrift | volume =4| issue=1b |pages =57–62}}</ref><ref>{{Cite journal | last=Abraham, M. | year=1903 | title=[[s:de:Prinzipien der Dynamik des Elektrons (1903)|Prinzipien der Dynamik des Elektrons]]| journal=Annalen der Physik | volume =10|pages =105–179}}</ref>
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| Meanwhile, Lorentz (1899, 1904) extended his theory of electrons, assuming that an electron's charge was spread throughout its volume, and that in Kaufmann's experiment, its shape would be compressed in the direction of motion and would stay unchanged in the transverse directions. To Kaufmann's surprise, Lorentz could show that his model agreed with his experimental data as well. This model was further elaborated and perfected by [[Henri Poincaré]] (1905), so that Lorentz's theory was now in agreement with the [[principle of relativity]].<ref group=A>Miller (1981), pp. 55–67</ref><ref group=A>Janssen (2007), section 4</ref><ref>{{Citation|author=Lorentz, Hendrik Antoon|year=1904|title=[[s:Electromagnetic phenomena|Electromagnetic phenomena in a system moving with any velocity smaller than that of light]]|journal=Proceedings of the Royal Netherlands Academy of Arts and Sciences|volume=6|pages=809–831}}</ref><ref>{{Citation|author=Poincaré, Henri|year=1906|title=[[s:fr:Sur la dynamique de l’électron (juillet)|Sur la dynamique de l’électron]] ([[s:Translation:On the Dynamics of the Electron (July)|On the Dynamics of the Electron]])|journal=[[Circolo Matematico di Palermo#Publications|Rendiconti del Circolo matematico di Palermo]]|volume=21|pages=129–176|doi=10.1007/BF03013466}}</ref>
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| A similar theory was developed by [[Alfred Bucherer]] and [[Paul Langevin]] in 1904, with the difference that the total volume occupied by the deformed electron was assumed unchanged. It turned out that this theory's prediction was closer to Abraham's theory than to Lorentz's.<ref group=A>Janssen (2007), section 4</ref><ref>A.H. Bucherer, Mathematische Einführung in die Elektronentheorie, Teubner, Leipzig 1904, p. 57</ref>
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| Finally, [[Albert Einstein]]'s theory of [[special relativity]] (1905) predicted the change of the [[point (geometry)|point-like]] electron's mass due to the properties of the transformation between the rest-frame of the particle and the laboratory frame in which the measurements were performed. Mathematically, this calculation predicts the same dependence between velocity and mass as Lorentz's theory, although it assumes very different physical concepts.<ref group=A>Staley (2008), pp. 241–242</ref><ref>{{Citation|doi=10.1002/andp.19053221004|author=Einstein, Albert|year=1905|title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik|volume=322|issue=10|pages=891–921|bibcode = 1905AnP...322..891E }}. See also: [http://www.fourmilab.ch/etexts/einstein/specrel/ English translation].</ref>
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| As regards the increase of transverse [[electromagnetic mass]], the predictions of the various theories were (Fig. 3):
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| :<math>\begin{align}
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| &\text{Abraham} & \phi(\beta) &=\frac{3}{4\beta^{2}}\left(\frac{1+\beta^{2}}{2\beta}\lg\frac{1+\beta}{1-\beta}-1\right)\\
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| &\text{Lorentz--Einstein} & \phi(\beta) &=(1-\beta^{2})^{-1/2}\\
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| &\text{Bucherer--Langevin} & \phi(\beta) &=(1-\beta^{2})^{-1/3}
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| \end{align}</math>
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| ===Experiments of 1905===
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| In order to make a decision between those theories, Kaufmann again performed his experiments with higher precision. Kaufmann believed that he had conclusively disproven the formula of Lorentz–Einstein, and had therefore also disproven the [[principle of relativity]]. In his view, the only remaining options were between the theories of Abraham and Bucherer. Lorentz was perplexed and wrote that he was "''at the end of his Latin''".<ref group=A>Miller (1981), pp. 228–232</ref><ref group=A>Staley (2008), pp. 242–244</ref><ref>{{Citation | author=Kaufmann, W. | year=1905 | title=Über die Konstitution des Elektrons ([[s:Translation:On the Constitution of the Electron (1905)|On the Constitution of the Electron]]) | journal=Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften | issue=45 |pages =949–956}}</ref><ref>{{Citation | author=Kaufmann, W. | year=1906 | title=Über die Konstitution des Elektrons ([[s:Translation:On the Constitution of the Electron (1906)|On the Constitution of the Electron]]) | journal=Annalen der Physik | volume =19 |pages =487–553}}</ref>
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| However, criticism of Kaufmann's experiment arose.<ref group=A>Miller (1981), pp. 232–235</ref><ref group=A>Staley (2008), pp. 244–250</ref> Shortly after Kaufmann published his results and the conclusions of his analysis, [[Max Planck]] decided to re-analyze the data obtained by the experiment. In 1906 and 1907, Planck published his own conclusion on the behavior of the inertial mass of electrons with high speeds. Using just nine data points from Kaufmann's publication in 1905, he recalculated the exact setup of the fields for each point, and compared the measurements against the predictions of the two competing theories. He showed that Kaufmann's results are not fully decisive and would lead to superluminal velocities.<ref>{{Citation|author=Planck, Max|year=1906|title=Die Kaufmannschen Messungen der Ablenkbarkeit der β-Strahlen in ihrer Bedeutung für die Dynamik der Elektronen ([[s:Translation:The Measurements of Kaufmann|The Measurements of Kaufmann on the Deflectability of β-Rays in their Importance for the Dynamics of the Electrons]])|journal=Physikalische Zeitschrift|volume=7|pages=753–761}}</ref><ref>M. Planck, ''Nachtrag zu der Besprechung der Kaufmannschen Ablenkungsmessungen'', Verhandlungen der Deutschen Physikalischen Gesellschaft, '''9''' (1907)</ref>
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| Einstein remarked in 1907 that although Kaufmann's results were better in agreement with Abraham's and Bucherer's theories than with his own, the foundations of the other theories were not plausible and therefore had only little probability of being correct.<ref>{{Citation|author=Einstein, Albert|year=1908
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| |title=Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen|journal=Jahrbuch der Radioaktivität und Elektronik|volume=4|pages=411–462|bibcode = 1908JRE.....4..411E }}</ref>
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| ==Subsequent experiments==
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| ===Bucherer===
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| {{double image
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| |6=Figure 4. Top view. Bucherer's experimental setup.
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| |7=Figure 5. Cross-section through the axis of the circular capacitor at angle α with respect to the magnetic field H.
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| The main problem of Kaufmann's experiments was his use of ''parallel'' magnetic and electric fields, as pointed out by [[Adolf Bestelmeyer]] (1907). Using a method based on ''perpendicular'' magnetic and electric fields (introduced by [[J. J. Thomson]] and further developed to a [[Wien filter|velocity filter]] by [[Wilhelm Wien]]), Bestelmeyer obtained considerably different values for the charge-to-mass ratio for cathode rays up to 0.3c. However, Bestelmeyer added that his experiment was not precise enough to provide a definite decision between the theories.<ref>{{Cite journal|author=Bestelmeyer, A.|year=1907|title= Spezifische Ladung und Geschwindigkeit der durch Röntgenstrahlen erzeugten Kathodenstrahlen|journal=Annalen der Physik|volume=327|issue=3|pages=429–447|doi=10.1002/andp.19073270303|bibcode = 1907AnP...327..429B }}</ref>
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| Therefore, [[Alfred Bucherer]] (1908) conducted a precise measurement using a velocity filter similar to Bestelmeyer's. See Figs. 3 & 5. A radium beta source was placed at the center of a circular condenser consisting of two silvered glass plates spaced 0.25 mm apart and charged to about 500 volts, set in a homogeneous 140 Gauss magnetic field. The radium emitted beta rays in all directions, but in any particular direction α, only those beta rays exited the velocity filter whose speed was such that the electric and magnetic fields exactly compensated each other. After leaving the condenser, the rays were deflected by the magnetic field, and exposed a photographic plate set parallel to the condenser rim and perpendicular to the non-deflected rays.<ref name=buch1>{{Citation|author=Bucherer, A. H.|year=1908|title=[[s:de:Messungen an Becquerelstrahlen|Messungen an Becquerelstrahlen. Die experimentelle Bestätigung der Lorentz-Einsteinschen Theorie.]] ([[s:en:Translation:Measurements of Becquerel rays|Measurements of Becquerel rays. The Experimental Confirmation of the Lorentz-Einstein Theory]])|journal=Physikalische Zeitschrift|volume=9|issue=22|pages=755–762}}</ref><ref name=buch2>{{Cite journal|author=Bucherer, A. H.|year=1909|title=[[s:de:Die experimentelle Bestätigung des Relativitätsprinzips|Die experimentelle Bestätigung des Relativitätsprinzips]]|journal=Annalen der Physik|volume=333|issue=3|pages= 513–536|doi=10.1002/andp.19093330305|bibcode=1909AnP...333..513B}}</ref>
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| {{Multiple image
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| |image1=BuchererExpGraph.svg
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| |caption1=Figure 6. Bucherer's data in five runs.
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| |alt1=Figure 6. Bucherer's data in five runs.
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| |image2=WolzExpGraph.svg
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| |caption2=Figure 7. Wolz's data in 13 runs.
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| |alt2=Figure 7. Wolz's data in 13 runs.
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| }}
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| For his final analysis, Bucherer recalculated the measured values of five runs with Lorentz's and Abraham's formulas respectively, in order to obtain the charge-to-mass ratio as if the electrons were at rest. Since the ratio doesn't vary for resting electrons, the data points should be on a single horizontal line (see Fig. 6). However, this was approximately only in the case when the data were calculated with Lorentz's formula, while the results of Abraham's formula sharply deviated (the red and blue lines represent the average value according to both formulas). The agreement with the Lorentz–Einstein formula was interpreted by Bucherer as the confirmation of the relativity principle and the Lorentz–Einstein theory – a result immediately applauded by Lorentz, Einstein, and [[Hermann Minkowski]].<ref group=A>Miller (1981), pp. 345–350</ref><ref group=A>Staley (2008), pp. 250–254</ref>
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| In addition, Bucherer's apparatus was improved in 1909 by his student [[Kurt Wolz]], who also obtained agreement with the Lorentz–Einstein formula (though he didn't compare Abraham's formula with his data, Fig. 7).<ref>{{Cite journal|author=Wolz, Kurt|year=1909|title= Die Bestimmung von e/m0|journal=Annalen der Physik|volume=335|issue=12|pages= 273–288|doi= 10.1002/andp.19093351206|bibcode = 1909AnP...335..273W }}</ref>
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| Even though many physicists accepted Bucherer's result, there still remained some doubts.<ref group=A>Miller (1981), pp. 345–350</ref><ref group=A>Staley (2008), pp. 250–254</ref> For instance, Bestelmeyer published a paper in which he cast doubt on the validity of Bucherer's result. He argued that one experiment alone cannot establish the correctness of an important physical law, that Bucherer's result might be significantly distorted by non-compensated rays reaching the photographic plate, and that extensive data protocols and error analysis are necessary.<ref>{{Cite journal|author=Bestelmeyer, A. H.|year=1909|title=Bemerkungen zu der Abhandlung Hrn. A. H. Bucherers: Die experimentelle Bestätigung des Relativitätsprinzips |journal=Annalen der Physik|volume=335|issue=11|pages=166–174|doi=10.1002/andp.19093351105|bibcode = 1909AnP...335..166B }}</ref> A polemic dispute between those two scholars followed in a series of publications, in which Bestelmeyer argued that Wolz's experiments are affected by the same problems.<ref>{{Cite journal|author=Bucherer, A. H.|year=1909|title=Antwort auf die Kritik des Hrn. E. Bestelmeyer bezüglich meiner experimentellen Bestätigung des Relativitätsprinzips |journal=Annalen der Physik|volume=335|issue=11|pages=974–986|doi=10.1002/andp.19093351506|bibcode = 1909AnP...335..974B }}</ref><ref>{{Cite journal|author=Bestelmeyer, A. H.|year=1910|title=Erwiderung auf die Antwort des Hrn. A. H. Bucherer|journal=Annalen der Physik|volume=337|issue=6|pages=231–235|doi=10.1002/andp.19103370609|bibcode = 1910AnP...337..231B }}</ref><ref>{{Cite journal|author=Bucherer, A. H.|year=1910|title=Erwiderung auf die Bemerkungen des Hrn. A. Bestelmeyer |journal=Annalen der Physik|volume=338|issue=14|pages=853–856|doi=10.1002/andp.19103381414|bibcode = 1910AnP...338..853B }}</ref>
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| === Hupka ===
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| Unlike Kaufmann and Bucherer, [[Karl Erich Hupka]] (1909) used cathode rays at 0.5c for his measurements. The radiation (generated at a copper cathode) was strongly accelerated by the field between cathode and anode in a highly evacuated discharge tube. The anode serving as a diaphragm was passed by the ray with constant velocity and drew the shadow image of two [[Wollaston wire]]s on a phosphorescent screen behind a second diaphragm. If a current was generated behind this diaphragm, then the ray was deflected and the shadow image was displaced. The results were in agreement with Lorentz–Einstein, even though Hupka remarked that this experiment didn't represent a definitive result.<ref>{{Cite journal|author=Hupka, E.|year=1910|title=Beitrag zur Kenntnis der trägen Masse bewegter Elektronen|journal=Annalen der Physik|volume=336|issue=1|pages= 169–204|doi=10.1002/andp.19093360109|bibcode = 1909AnP...336..169H }}</ref> Subsequently, W. Heil published some papers dealing with criticisms and interpretations of the result, to which Hupka replied.<ref>{{Cite journal|author=Heil, W.|year=1910|title=Diskussion der Versuche über die träge Masse bewegter Elektronen|journal=Annalen der Physik|volume=336|issue=3|pages= 519–546|doi=10.1002/andp.19103360305|bibcode = 1910AnP...336..519H }}</ref><ref>{{Cite journal|author=Hupka, E.|year=1910|title=Zur Frage der trägen Masse bewegter Elektronen |journal=Annalen der Physik|volume=338|issue=12|pages= 400–402|doi=10.1002/andp.19103360305|bibcode = 1910AnP...336..519H }}</ref><ref>{{Cite journal|author=Heil, W.|year=1910|title=Zur Diskussion der Hupkaschen Versuche über die träge Masse bewegter Elektronen |journal=Annalen der Physik|volume=338|issue=12|pages= 403–413|doi=10.1002/andp.19103381210|bibcode = 1910AnP...338..403H }}</ref>
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| === Neumann and Guye/Lavanchy===
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| [[File:NeumannExpGraph.svg|thumb|left|290px|Figure 8. Neumann's evaluation of 26 data points for each theory.]]
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| In 1914, [[Günther Neumann]] conducted new measurements using Bucherer's equipment, in particular making some improvements to address Bestelmeyer's criticisms, especially the question of non-compensated rays, and making extensive refinements to the data protocols. The calculation method was the same as Bucherer's (see Fig. 6). Also in this experiment, the data corresponding to Lorentz's formula are nearly on a horizontal line as required, while the data obtained from Abraham's formula sharply deviate (see Fig. 8). Neumann concluded that his experiments were in agreement with those of Bucherer and Hupka, definitely proving the Lorentz–Einstein formula in the range 0.4–0.7c, and refuted Abraham's formula. Instrumental uncertainties occurred in the range 0.7–0.8c, so the deviation from the Lorentz–Einstein formula in this range wasn't considered as being significant.<ref>{{Cite journal|author=Neumann, Günther|year=1914|title=Die träge Masse schnell bewegter Elektronen|journal=Annalen der Physik|volume=350|issue=20|pages= 529–579|doi=10.1002/andp.19143502005|bibcode=1914AnP...350..529N}}</ref>
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| [[File:GuyeLavanchyGraph.svg|thumb|right|290px|Figure 9. Guye and Lavanchy's evaluation of 25 data points for each theory.]]
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| In 1915, [[Charles-Eugène Guye]] and [[Charles Lavanchy]] measured the deflection of cathode rays at 0.25c–0.5c. They used a tube with a cathode and anode in order to accelerate the rays. A diaphragm at the anode produced a beam which was deflected. A screen was placed at the end of the apparatus, at which the impacts were photographed by a camera. They subsequently computated the ratio of transverse electromagnetic mass ''m<sub>T</sub>'' and rest mass ''m<sub>0</sub>'' indicated by the red and blue curve, and obtained good agreement with the Lorentz–Einstein formula (see Fig. 9), supplementing Neumann's result.<ref>{{Cite journal|author=C.E. Guye, C. Lavanchy|year=1915|title=Vérification expérimentale de la formule de Lorentz–Einstein par les rayons cathodiques de grande vitesse|journal=Compt. Rend. Acad. Sci.|volume=161|pages= 52–55|url=http://gallica.bnf.fr/ark:/12148/bpt6k3114b/f52}}</ref><ref>{{Cite journal|author=C.E. Guye, C. Lavanchy|year=1915|title=Vérification expérimentale de la formule de Lorentz–Einstein par les rayons cathodiques de grande vitesse|journal=Archives des sciences physiques et naturelles|volume=42|pages= 286ff|url=http://www.archive.org/details/archivesdesscience42genv}}</ref>
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| Neumann's and Guye/Lavanchy's experiments were considered by many as conclusively proving the Lorentz-Einstein formula.<ref group=A>Pauli (1921), p. 636</ref><ref group=A>Miller (1981), pp. 350-351</ref><ref group=A>Staley (2008), pp. 254–257</ref> Lorentz summarized these efforts in 1915:<ref group=A>Lorentz (1915), p. 339</ref>
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| :''Later experiments [..] have confirmed the formula [..] for the transverse electromagnetic mass, so that, in all probability, the only objection that could be raised against the hypothesis of the deformable electron and the principle of relativity has now been removed.''
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| == Further development ==
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| [[File:Rogers et al electrostatic spectrograph (1940).svg|thumb|left|275px|Figure 10. Rogers ''et al.'' electrostatic spectrograph]]
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| Zahn & Spees (1938)<ref>{{Citation | author=Zahn, C. T. and Spees, A. A. | year=1938 | title= A Critical Analysis of the Classical Experiments on the Variation of Electron Mass | journal=Physical Review | volume=53| pages= 511–521 | doi=10.1103/PhysRev.53.511 | bibcode=1938PhRv...53..511Z}}</ref> and Faragó & [[Lajos Jánossy]] (1954)<ref name=farago>{{Citation | author=P. S. Faragó and L. Jánossy | year=1957 | title= Review of the experimental evidence for the law of variation of the electron mass with velocity | journal=Il Nuovo Cimento |volume=5| issue=6|pages= 379–383 | doi=10.1007/BF02856033}}</ref> argued that many assumptions employed in those early experiments as to the nature and the properties of electrons and the experimental setup, were wrong or imprecise. As with Kaufmann's experiments, the Bucherer–Neumann experiments would only show a qualitative increase of mass, and were incapable of deciding between the competing theories.<ref group=A>Miller (1981), pp. 351–352</ref><ref group=A>Janssen (2007), section 7</ref>
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| While the results of those electron deflection experiments were disputed for a long time, the investigations of the [[fine structure]] of the [[Balmer series|hydrogen lines]] by [[Karl Glitscher]] (based on the work of [[Arnold Sommerfeld]]) had already in 1917 provided a clear confirmation of the Lorentz–Einstein formula, because the relativistic expressions for momentum and energy were necessary to derive the fine structure, and a constituted a refutation of Abraham's theory.<ref>{{Cite journal|author=Glitscher, Karl|year=1917|title= Spektroskopischer Vergleich zwischen den Theorien des starren und des deformierbaren Elektrons|journal=Annalen der Physik|volume=357|issue=6|pages= 608–630|doi=10.1002/andp.19173570603|bibcode = 1917AnP...357..608G }}</ref><ref group=A>Pauli (1921), pp. 636–637</ref>
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| [[File:RogersExpGraph.svg|thumb|300px|Figure 11. Three data points of Rogers ''et al''., in agreement with the Lorentz–Einstein formula.]]
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| In addition, the first electron deflection experiments with sufficient precision was conducted by Rogers ''et al.'' (1940), who developed an improved setup. The [[Decay chain|radium decay series]] yields a spectrum of beta particles with a wide range of energies. The earlier measurements by Kaufmann, Bucherer, and others had used flat parallel plate condensers which provided no focusing of the beta particles. Rogers ''et al.'' (Fig. 10) instead constructed an electrostatic spectrograph capable of resolving the energy maxima of individual beta particle lines from the radium decay series. The electrostatic spectrograph was constructed of segments of two cylinders and was enclosed in an evacuated iron box. The beta rays were emitted from a fine platinum wire coated with radium active-deposit. The dispersed rays were incident on a slit in front of a [[Geiger counter]]. The data from this experiment was combined with previous magnetic spectrometer measurements of ''H<sub>ρ</sub>'' to yield the charge-to-mass ratio, which was subsequently compared with the predictions of Lorentz and Abraham for the ratio of transverse mass and rest mass. The points were all on the curve representing the Lorentz–Einstein formula to within 1% (see Fig. 11).<ref>{{Citation | author=Rogers, M. M. et al. | year=1940 | title= A Determination of the Masses and Velocities of Three Radium B Beta-Particles | journal=Physical Review |volume=57| pages= 379–383 | doi=10.1103/PhysRev.57.379 | bibcode=1940PhRv...57..379R}}</ref> This experiment is seen as being sufficiently precise to distinguish between the theories.<ref group=A>Janssen (2007), section 7</ref>
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| ==Modern Tests==
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| {{Main|Tests of relativistic energy and momentum}}
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| Since then, many additional experiments concerning the relativistic mass-energy-momentum relation have been conducted, including measurements of the deflection of electrons, all of them confirming special relativity to high precision. Also in modern [[particle accelerator]]s, the predictions of special relativity are routinely confirmed.
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| ==See also==
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| *[[History of special relativity]]
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| ==References==
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| === Primary sources ===
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| {{Reflist|2}}
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| === Secondary sources ===
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| *{{Cite journal|author=Battimelli, G.|year=1981|title=The Electromagnetic Mass of the Electron: A Case Study of a Non-Crucial Experiment|journal=Fundamenta Scientiae|volume=2|pages=137–150}}
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| *{{Citation
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| |author=Janssen, Michel & Mecklenburg, Matthew
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| |year=2007
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| |contribution=From classical to relativistic mechanics: Electromagnetic models of the electron
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| |editor=V. F. Hendricks, et al.
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| |title=Interactions: Mathematics, Physics and Philosophy
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| |pages= 65–134
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| |location=Dordrecht
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| |publisher=Springer
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| |contribution-url=http://philsci-archive.pitt.edu/archive/00001990/}}
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| *{{Citation
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| |author=Lorentz, Hendrik Antoon
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| |year=1916
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| |title={{Internet archive|electronstheory00lorerich|The theory of electrons and its applications to the phenomena of light and radiant heat}}
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| |location=Leipzig & Berlin
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| |publisher=B.G. Teubner}}
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| *{{Citation
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| |author=Miller, Arthur I.
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| |year=1981
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| |title= Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911)
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| |location= Reading
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| |publisher=Addison–Wesley
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| |isbn=0-201-04679-2}}
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| *{{Citation
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| |author=Pais, Abraham
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| |authorlink=Abraham Pais
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| |year=1982/2005
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| |title= Subtle is the Lord: The Science and the Life of Albert Einstein
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| |location= New York
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| |publisher=Oxford University Press
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| |isbn=0-19-280672-6}}
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| *{{Citation
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| |author=Pauli, Wolfgang
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| |authorlink=Wolfgang Pauli
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| |year=1921
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| |journal=Encyclopädie der mathematischen Wissenschaften
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| |title= Die Relativitätstheorie
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| |pages=539–776
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| |volume=5
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| |issue=2
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| |url=http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=201990}}; In English: {{cite book|last=Pauli|first=Wolfgang|title=Theory of Relativity|publisher=Dover|location=New York|year=1981|isbn=0-486-64152-X}}
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| *{{Citation
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| |author=Staley, Richard
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| |year=2008
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| |title=Einstein's generation
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| |publisher=University Press
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| |location=Chicago
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| |isbn=0-226-77057-5}}
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| {{Reflist|2|group=A}}
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| ==External links==
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| *[http://www.nd.edu/~dhoward1/The%20Kaufmann%20Experiemnts.ppt Presentation on Kaufmann experiment] (PowerPoint)
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| *[http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html List of SR tests]
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| {{Tests of special relativity}}
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| {{DEFAULTSORT:Kaufmann-Bucherer-Neumann experiments}}
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| [[Category:Physics experiments]]
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| [[Category:Special relativity]]
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