Experimental Confirmation of the Lorentz—Einstein Formulas.

S. E. Frisch.

Both classical electrodynamics and the principle of relativity lead to a dependence of the electron mass on its velocity. In both cases

\[ m_v=m_0\varphi(\beta), \]

where \(m_v\) is the mass of the electron at velocity \(v\), \(m_0\) is the mass at infinitely small velocity, and \(\beta=\dfrac{v}{c}\), where \(c\) is the speed of light. However, the form of the function \(\varphi\) will be different in the two cases.

In 1903 Abraham\(^{1}\), assuming that the electron is an incompressible sphere with constant charge density and using the basic principles of electrodynamics, determined the form of the function \(\varphi\).\(^{2}\) It is necessary to distinguish two masses: the longitudinal, or kinetic, mass \(m_t\), and the transverse mass \(m_n\), which appears under a force acting normal to the trajectory. Abraham obtained:

\[ m_n=m_0\frac{3}{4\beta^2}\left[\frac{1+\beta^2}{2\beta}\log\frac{1+\beta}{1-\beta}-1\right]\ldots\ldots (1a) \]

\[ m_t=m_0\frac{3}{4\beta^2}\left[\frac{2}{1-\beta^2}-\frac{1}{\beta}\log\frac{1+\beta}{1-\beta}\right]\ldots\ldots (1b) \]

Lorentz\(^{3}\), wishing to explain the negative result of the Michelson and Morley experiments, proposed that the electron is flattened during motion. The radius \(R\) in the direction of motion is transformed into \(R\sqrt{1-\beta^2}\), while the radius perpendicular to the direction of motion retains its magnitude. In this case the dependence of the mass on velocity is obtained as follows:

\[ m_n=m_0(1-\beta^2)^{-\frac{3}{2}}\ldots\ldots\ldots\ldots (2a) \]

\[ m_n=m_0(1-\beta^2)^{-\frac{1}{2}}\ldots\ldots\ldots\ldots (2b) \]

\(^{1}\) Abraham, Ann. d Phys. 10, p. 105, 1903.

\(^{2}\) The value of \(m_0\) depends on whether we regard the electron as a sphere with volume or surface charge, but the form of the function \(\varphi(\beta)\) does not depend on this.

\(^{3}\) Lorentz, Proceedings Acad. Sc. Amsterdam, 6.

Einstein’s principle of relativity1) leads to the same dependence of mass on velocity as Lorentz’s theory.

It is for experiment to decide which of the formulas corresponds to reality. By deflecting a beam of electrons once in an electric field and another time in a magnetic field, one can determine the velocity of the electrons \(v\) and the ratio of charge to mass \(\frac{\epsilon}{m}\). Taking the electron charge \(\epsilon\) to be a constant quantity, one can find the mass corresponding to the given velocity.

Since in experiment we are dealing with the deflection of electrons, and not with a change in their velocity, the transverse mass enters into the formulas. Thus the theoretical formulas \((1a)\) and \((2a)\) are subject to experimental verification. Expanding these formulas in series, we obtain:

\[ m_n = m_0 \left\{ 1 + \frac{6}{3.5}\beta^2 + \frac{9}{5.7}\beta^4 + \cdots \right\} \ldots \quad (1c) \]

\[ m_n = m_0 \left\{ 1 + \frac{1}{2}\beta^2 + \frac{3}{8}\beta^4 + \cdots \right\} \ldots \quad (2c) \]

The formulas differ only in the terms with \(\beta^2\) and higher orders, whence follows the difficulty of experimentally verifying which of them is correct.

In view of the importance of the question, a large number of experimental investigations were undertaken. Some authors worked with the \(\beta\)-rays of radium, advantageous because of their great speed; others, with cathode rays.

Kauffmann2) from 1902 to 1906 carried out extensive work with the \(\beta\)-rays of radium. He deflected a beam of rays simultaneously in an electric and a magnetic field in two perpendicular directions. After deflection the beam fell upon a photographic plate. Owing to the presence in the beam of electrons of different velocities, a continuous curve was obtained on the plate. By changing the direction of the electric field, Kauffmann obtained a second curve, symmetric with the first. The coordinates of these curves were measured under a microscope. Knowing these coordinates and the intensities of the deflecting fields, one can determine the dependence of mass on velocity. The results obtained confirmed Abraham’s theory and did not agree with the requirements of the Lorentz–Einstein theory. Kauffmann’s work was subjected to careful analysis by Planck, Stark, and Heil. Heil3) showed that, owing to an error of 4% in the determination of the electric-field voltage, the results cannot be regarded as confirming Abraham’s theory.

1) Einstein, Ann. d. Phys. 17, p. 891, 1905.
2) Kauffmann, Ann. d. Phys. 19, p. 447, 1906.
3) Heil, Dissertation. Berlin. 1909.

In 1908 Bucherer1 undertook a new investigation with \(\beta\)-rays, using another method. First the beam of \(\beta\)-rays entered a region where it was acted upon simultaneously by an electric and a magnetic field, directed in such a way and having such an intensity that they compensated one another; here the beam retained its rectilinear direction. Then it was subjected to the action of a single magnetic field, under whose influence it described part of a circumference and, finally, struck a photographic plate. Measuring the deflection and knowing the fields, it was possible to determine \(\beta\) and, from the theoretical formulae, to calculate the corresponding value of \(\dfrac{e}{m_0}\). The correct formula would be the one which gives a constant value for \(\dfrac{e}{m_0}\). Bucherer found that his work confirmed the formulae of Lorentz–Einstein.

Bestelmeyer2 subjected Bucherer’s work to criticism. Neumann3 repeated the work by Bucherer’s method and obtained still better agreement with the Lorentz–Einstein theory.

The following investigations were undertaken with cathode rays. In 1910 Proctor4 deflected a beam of cathode rays of high velocity in electric and magnetic fields. For the region from \(\beta = 0.12\) to \(\beta = 0.43\) he obtained results agreeing with Abraham’s theory.

In the same year Hupka5 carried out a new investigation with cathode rays. He determined the velocity of the electrons by measuring the discharge potential, and, by deflecting the beam in a known magnetic field, found \(\dfrac{e}{m}\). Hupka obtained agreement with Lorentz–Einstein’s formula, but his work was subjected to serious criticism by Heil.

Thus the investigations cited above gave highly contradictory results. This is explained by the great difficulty of the measurements. As is readily seen, the deflection of an electron in an electric field will be

\[ x = A\,\frac{e}{m}\,\frac{1}{v^2}\,V \quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \quad (3) \]

and in a magnetic field:

\[ y = B\,\frac{e}{m}\,\frac{1}{v}\,J \quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \quad (4) \]

Here \(V\) is the potential difference on the plates of the capacitor producing the electric field, \(J\) is the current strength in the solenoid producing the magnet—

field (we assume that the magnetic field is produced not by an electromagnet, but by a solenoid without a core). \(A\) and \(B\) are expressed by the formulas:

\[ A=\int_0^{v_0} dl \int_0^l F_1\,dl;\quad B=\int_0^{i_0} dl \int_0^l H_1\,dl \qquad \ldots\ldots\ldots (5), \]

where \(F_1\) and \(H_1\) are the values of the electric and magnetic fields for \(V=1\) and \(J=1\). In view of the incomplete homogeneity of the fields, \(A\) and \(B\) will be different for different trajectories; they will depend on the deflection of the beam. The impossibility of determining \(A\) and \(B\) exactly is the chief source of error for all observers.

To avoid the error arising from the nonconstancy of \(A\) and \(B\), Guye and Ratnowsky1 undertook a study in which they each time brought the deflection of the beam to one and the same magnitude. In this way the electrons always moved along the same trajectories, and \(A\) and \(B\) remained constant. The study confirmed the Lorentz–Einstein formula.

Considering that, for the solution of so important a question as the dependence of mass on velocity, a single study carried out by the correct method was insufficient, Guye, together with Lavanchy, repeated the work with particular care. The present abstract is devoted to the description of this work.2

Guye and Lavanchy call their method the method of “coincident trajectories.” As already stated, it consists in bringing the deflection of the beam of electrons each time to one and the same magnitude. In order for the electrons to move in the magnetic field along one and the same trajectory, it is sufficient that

\[ \frac{J}{mv}=\frac{J'}{m'v'} \qquad \ldots\ldots\ldots\ldots\ldots\ldots (6), \]

where \(m\) and \(m'\) are the masses of electrons of velocities \(v\) and \(v'\), and \(J\) and \(J'\) are the strengths of the currents in the solenoids producing the magnetic field. In the magnetic field the force is normal to the trajectory; therefore the electrons, during the entire time of their motion in the magnetic field, do not experience tangential acceleration, and their velocity \(v\) and mass \(m\) remain unchanged. In the electric field the force is not always normal to the trajectory, but calculation shows that the change in the velocity \(v\) is much smaller than the observational error, so that it may be neglected. Then the condition that the electrons move in the electric field along one and the same path will be:

\[ \frac{V}{mv^2}=\frac{V'}{m'v'^2} \qquad \ldots\ldots\ldots\ldots\ldots (7), \]

where \(V\) and \(V'\) are the voltages on the plates of the capacitor. From formulas (6) and (7) we obtain:

\[ \frac{v}{v'}=\frac{J V'}{J' V}. \tag{8} \]

\[ \frac{m}{m'}=\frac{V J'^2}{V' J^2}. \tag{9} \]

The ratios \(\frac{m}{m'}\) and \(\frac{v}{v'}\) are thus obtained by comparing the relative values of two current intensities and two potential differences. To compare the results with theory it is necessary to know the absolute value of only one velocity \(v\)—the “comparison velocity”—and the corresponding \(\frac{m}{m_0}\). The determination of the “comparison velocity” is facilitated by the fact that this velocity may be taken small. The method of determining the “comparison velocity” will be indicated below. The ratio \(\frac{m}{m_0}\) corresponding to the “comparison velocity” is calculated from the theoretical formulas.

A drawback of the method is that the need to bring the deflection accurately each time to one and the same value does not allow one to make observations rapidly one after another. The latter is necessary for electrons of high velocity, in view of the instability of the discharge-tube regime. To eliminate this drawback a photographic method of observation was introduced, and the deflection was each time brought not exactly, but approximately, to the previous value. Specially undertaken experiments showed that for nearby trajectories the integral actions of the fields \(A\) and \(B\) change little, so that they may be taken as equal. Suppose that for electrons of velocity \(v\) we have the electric deflection \(x_o\) and the magnetic deflection \(y_o\), and for electrons of velocity \(v'\)—the deflections \(x_n\) and \(y_n\); then

\[ x_o=A_o\,\frac{e}{m}\,\frac{V}{v^2};\qquad x_n=A_n\,\frac{e}{m'}\,\frac{V'}{v'^2} \]

\[ y_o=B_o\,\frac{e}{m}\,\frac{J}{v};\qquad y_n=B_n\,\frac{e}{m'}\,\frac{J}{v'}, \]

whence:

\[ \frac{v'}{v}=\frac{K}{L}\,\frac{J\cdot V'}{V\cdot J'}\cdot \frac{x_o\cdot y_n}{y_o\cdot x_n} \]

\[ \frac{m'}{m}=\frac{L^2}{K}\, \frac{V\cdot J'^2\cdot y_o^2\cdot x_n}{J^2\cdot V'\cdot x_o y_n^2}, \]

where

\[ K=\frac{A_o}{A_n},\quad L=\frac{B_o}{B_n}; \]

putting \(A_o=A_n\), \(B_o=B_n\), we obtain:

\[ \frac{v'}{v}=\frac{J V'}{J' V}\cdot\frac{x_o\cdot y_n}{y_o\cdot x_n} \tag{10} \]

\[ \frac{m'}{m}=\frac{V J'^2}{V' J^2}\cdot\frac{y_o^2 x'}{x\,y_n^2}. \tag{11} \]

For \(x_o=x_n,\ y_o=y_n\) we obtain the previous formulas (8) and (9).

Introducing the quantities \(\beta=\dfrac{v}{c}\) and \(\dfrac{m}{m_0}\), we may rewrite formulas (10) and (11) as:

\[ \beta'=\left[\beta\,\frac{Jx}{Vy}\right]\frac{V'y'}{J'x'} \tag{10a} \]

\[ \frac{m'}{m_0}=\left[\frac{m}{m_0}\frac{Vy^2}{J^2x}\right]\frac{J'^2x'}{V'y'^2} \tag{11a} \]

Here the primed quantities refer to the given beam of electrons, and the unprimed ones to the slow “comparison beam.” The unprimed quantities are determined once. The advantages of the method are seen in the final formulas (10a) and (11a), from which it is evident that the absolute value \(\beta'\) and \(\dfrac{m'}{m_0}\) depends only on a single absolute value—the “comparison velocity” \(\beta\) in the slow beam. All experimental quantities \(J, J', V, V', x, x', y, y'\) enter the formulas in such a way that \(\beta'\) and \(\dfrac{m'}{m_0}\) depend only on their ratios. Since \(J, J'\) and \(V, V'\) are measured all the time with one and the same instrument, and \(x, x'\) and \(y, y'\) with one and the same ruler, the systematic errors must be very small. Random errors are eliminated by a large number of observations.

The cathode tube and the general arrangement of the apparatus are shown in drawing 1. The source of current was a Wimshurst machine \(M\) with 8 plates, driven by an electric motor. The machine gave sufficient voltage to obtain electrons with velocity \(\tfrac12 c\). The cathode beam obtained was very homogeneous. The voltage was regulated by the number of sparks \(B\) and by the distance between them. To obtain a steady discharge for each voltage it was necessary to select the most suitable vacuum. The rarefaction was produced by a Gaede pump. When the vacuum had been selected, the stopcock was closed.

Fig. 1.

The total length of the cathode tube was about 80 cm; the diameter was 3 cm near the cathode and 8 cm at the end. For convenience of disassembly and cleaning, the tube consisted of two parts joined by sealing wax. The cathode was made of aluminum. The anode was a brass cylinder connected to earth. The diaphragm \(a\)—with a round aperture of 0.2 mm. At the anode tube there was an enlargement supporting the condenser. The part of the tube between the anode and the screen was glued-

...by a stanniol sheet connected to earth. Owing to this, the cathode beam entered a Faraday cylinder, consisting of tubes \(B\) and \(D\) and a stanniol lining, and was thereby protected from the action of an external electric field. The end of the tube was closed by a screen with a fluorescent substance. The earth’s magnetic field was weakened by an iron frame; in addition, the tube was placed along the magnetic meridian, so that the earth’s field deflected the beam in the horizontal direction, while the vertical deflection was being measured. As is known, there is a dependence between the cathode–anode distance and the vacuum required for obtaining electrons of maximum velocity. The work was carried out with tubes of various sizes, which made it possible to obtain fast electrons easily. The deflecting electric field was produced by a condenser made of two brass plates \(pp\), of dimensions \(2.4\ \text{cm} \times 5\ \text{cm}\). The plates were not plane, but were bent so that the form of the equipotential surfaces was close to the form of the trajectory. Moreover, this had a practical advantage: it was possible to obtain a strong deflection of the beam without forcing it to pass too close to the edges of the condenser. The potential was produced by a battery \(P\) of dry cells of the type used in pocket flashlights, in all 250 batteries of 3 cells each; the voltage was 1000 volts. The middle of the batteries was earthed, owing to which the charge of the condenser was symmetrical. A commutator made it possible to change the direction of the field. The magnetic field was produced by coils fed by current from a storage-battery \(Ac\). The coils caused a deflection of the beam in the vertical direction, i.e. in the same direction as the electric field.

\(V\) and \(J\) were measured with a Siemens and Halske milliammeter \(MA\). By inserting a resistance, it could be used as a voltmeter for measuring \(V\); by inserting the shunt \(S\), as an ammeter for measuring \(J\). The scale was carefully checked. \(V\) and \(J\) could be measured with an accuracy up to \(\tfrac{1}{4}\%\).

The bright spot on the screen was photographed with camera \(A\). Black crosses were marked on the edges of the screen. Before and after the exposure, the parts of the screen on which the crosses had been marked were illuminated by an electric lamp. On the negative two black strips with white crosses were obtained. They served as a scale for measurements and indicated that during the exposure there had been no displacement.

The deflections in the electric and magnetic fields are both vertical. They can be distinguished by making them merely unequal. The deflection in the magnetic field was always taken to be the stronger. Each observation consisted of two double deflections—the magnetic, once upward and once downward, and the same two deflections in the electric field. In addition, the undeflected beam was photographed. Thus five points were obtained on the photographic plate, situated on one vertical line. The middle one corresponded to the undeflected beam; the two outer ones to the two magnetic deflections, and the two intermediate ones to the two electric deflections.

Several (10–18) such photographs were made on each plate. The entire series of photographs on a single plate was taken at one and the same rotation speed of the electrostatic machine and in one and the same vacuum; thus the velocity of the electrons was one and the same. \(J\) and \(V\) remained unchanged throughout, and their values were measured at the beginning and at the end of the photographing. On the negatives, with the aid of a ruler graduated to \(0.1\ \mathrm{mm}\), the double electric and magnetic deflection was measured. The accuracy of the measurements was \(\frac{1}{300}\). Each measurement was taken as the mean of 10.

The absolute determination of the “comparison velocity” \(v\) presented great difficulties. It would seem that the simplest way to determine \(v\) is from the known relation:

\[ Ue=\frac{1}{2}m_t v^2 \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (12), \]

where \(U\) is the discharge potential. However, considerations of a practical nature compelled us to adopt another method. Formula (12), in conjunction with (3), gives:

\[ v=\sqrt{\frac{A}{x}\frac{e}{m_0}\frac{m_0}{m}V}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (13). \]

\[ A=2\frac{U}{V}\frac{m}{m_t}x \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (14). \]

It was by these formulas that \(v\) was determined. This method has the following advantages: first \(A\) is determined, and here all attention can be directed to the exact determination of \(U\) and \(V\), and there is no need to worry about keeping the electron velocity \(v\) strictly constant throughout the observations. Then the “comparison velocity” \(v\) itself is determined, together with the corresponding \(x, y, V\), and \(J\), which also enter into the final formulas (10a) and (11a). In this case it is no longer necessary to determine the discharge potential \(U\).

The discharge potential \(U\) was close to \(14000\ \mathrm{v}\). It was measured by an ordinary electrometer, which was carefully graduated (and for exactly the same volts) absolutely. The observational error with the electrometer was \(\frac{1}{150}\); the error of comparison with the absolute was \(\frac{1}{200}\). In view of the fact that the discharge potential \(U\) was measured only once and was comparatively not high, it could be determined with sufficient accuracy.

\(\frac{m}{m_t}\) and \(\frac{m_0}{m}\) were determined from theoretical formulas. At first they were taken equal to 1 and \(v\) was determined; for the computed \(v\), \(\frac{m}{m_t}\) and \(\frac{m_0}{m}\) were found, and the velocity was again determined, etc. Practically the second approximation always coincided with the third.

For \(\frac{e}{m_0}\) the value found by other observers was taken, namely \(1.77\cdot 10^7\).

The velocity of comparison \(v\) was obtained as the mean of 200 measurements. For the Lorentz–Einstein theory it was equal to:

\[ \beta = 0,2279 \]

and for Abraham’s theory:

\[ \beta = 0,2286. \]

In all, Guye and Lavanchy obtained 150 negatives, on which there were about 2000 double deflections. All observational and computational data were published in the Mémoires de la Société de Physique et d’Histoire naturelle de Genève. The results are collected in large tables, separately for the Lorentz–Einstein theory and separately for Abraham’s theory. Finally, Guye and Lavanchy give the final table, which we reproduce here:

Advances in Physical Science.

The same results are given graphically in Figure 2. The solid curve \(L-E\) is the theoretical Lorentz–Einstein curve; the curve \(A\) is Abraham’s curve. The points show the experimental data calculated according to the Lorentz–Einstein theory; the circles—according to Abraham’s theory. Accordingly, the comparison rays for the two theories are denoted by the symbols \(\odot\) and \(\bigcirc\).

Fig. 2.

From consideration of the table and the curves it follows without doubt that the experiment confirms the Lorentz–Einstein theory. The deviations from the theoretical curve are extremely small and have different signs. The mean deviation \(\Delta = +0.0002\), i.e. practically equal to zero. The deviations from Abraham’s curve are systematic: the calculated values are always smaller than the observed ones. The mean deviation \(\Delta = +0.0112\), i.e. much greater than the observational error.

The work of Guye and Lavanchy was carried out so carefully, and the agreement with theory is so striking, that at the present time the validity of the Lorentz–Einstein formula may be regarded as experimentally proven.

1. Guye et Ratnowsky, C. R. 150, p. 326, 1910; Arch. des Sc. Phys. et Nat. 31, p. 293, 1911. ↩↩

2. Ch. E. Guye et Lavanchy, Vérification expérimentale de la formule de Lorentz-Einstein par les rayons cathodiques de grande vitesse. Arch. des Sc. Phys et Nat. XLII, pp. 287, 353, 441; 1916. ↩↩

3. Neumann, Ann. d. Phys. 45, p. 529, 1914. ↩

4. Proctor, Phys. Rev. 30, p. 53, 1910. ↩

5. Hupka, Ann. d. Phys. 31, p. 169, 1910. ↩