Translation:The Electromagnetic Mass of the Electron

by Walter Kaufmann (physicist)

The electromagnetic mass of the electron.

W. Kaufmann (Göttingen)

At the meeting of natural scientists in Hamburg^[1], I was able to report about an experiment with the result that the ratio $\epsilon /\mu$ of Becquerel rays would decrease with increased velocity, and $\mu$ would increase if one assumes $\epsilon$ as constant, namely it increases the quicker, the more the velocity $(q)$ would approach the speed of light $(c)$. Such a behavior is theoretically given from the equation of energy of a quickly moving electric charge. At that time, it was achieved to bring the results into accordance with the theoretical formula derived by Searle^[2]; but only under the assumption, that the greatest part of the mass of the moving electron would be of mechanical origin, and the rest of electromagnetic origin. However, soon after publication of the experiments of that time, it was shown by Abraham,^[3], that Searle's formula for the field energy of the moving electron, only allows to calculate (without further ado) the electromagnetic mass in the case of acceleration in the direction of motion, while on the other hand, for the transverse acceleration (as it was the case in my experiments) an expression for mass applies which deviates from Searle's formula. If $\beta =q/c$, $\epsilon$ is the charge of the electron in E.M.E, $\mu _{0}$^[4] is the value of electromagnetic mass for small velocities, then it is according to Abraham:

1)	${\frac {\epsilon }{\mu }}={\frac {\epsilon }{\mu _{0}}}{\frac {4}{3}}{\frac {1}{\psi (\beta )}}$

where

2)	$\psi (\beta )={\frac {1}{\beta ^{2}}}\left[{\frac {1+\beta ^{2}}{2\beta }}\ln \left({\frac {1+\beta }{1-\beta }}\right)-1\right],$

($\psi (\beta )={\tfrac {4}{3}}$ for $\beta =0$; $\psi (\beta )=\infty$ for $\beta =1$).

A comparison of my experimental results with his formula that was already undertaken by Abraham, gave no good agreement, since the mass varies faster as it was required by the theory, so that a possibly added mechanical mass had to be set as negative.

In what follow, a more rational value for the evaluation of the results shall be shown, and at the same time by the aid of new experimental material, the full agreement between observation and theory shall be proven.

In the observation of that time, the absolute values of $q$ and $\epsilon \mu$ have been determined by using the absolute values of the electric and magnetic field, though already at that time, the possible errors were estimated to ca. 5%; these are errors which are much higher than the relative errors when measuring the plates.

Due to the large variability of $\psi (\beta )$ for $\beta$ nearly equal to 1, a small error of $\beta$ means a very large error for $\mu$ (for $\beta =0,96$ or 0,98, it is for instance $\psi (\beta )=3,141$, i.e., to an error of 2% in determining of $\beta$, an error of 19% for $\mu$ is corresponding.).

A rational utilization of the curve measured on the plate is only achieved by comparing the relative values with each other; one may not determine the curve constants directly by measuring the apparatus dimensions and the field, but one has to determine the most probable values by the method of least squares.

Let $y$ and $z$ be the magnetic deflections measured on the plate. From them, two other quantities $\eta$ and $\zeta$ can be derived, which have a simple relation to $\epsilon /\mu$ and $q$. $\eta$ and $\zeta$ are approximately proportional to $y$ and $z$; the deviations from proportionality can be represented by correction-terms depending on the apparatus dimensions, so that even considerable errors in the determination of the latter ones, are of little influence on the results.^[5]

Table I
z cm	y cm	β	ψ(β)	k2	δ perc.
0,348 0,461 0,576 0,688	0,0839 0,1175 0,1565 0,198	0,957 0,907 0,847 0,799	3,08 2,49 2,13 1,96	2,16 2,165 2,20 2,165 Mean: 2,173	-0,6 -0,4 +1,2 -0,4	k1=0,532 $\epsilon ={\sqrt {\frac {\Sigma \delta ^{2}}{3}}}=\pm \ 0,8$ perc.

Table II
z cm	y cm	β	ψ(β)	k2	δ perc.
0,200 0,250 0,300 0,350 0,400 0,450 0,525	0,0241 0,0305 0,0382 0,0469 0,0574 0,0688 0,0856	0,930 0,917 0,875 0,831 0,777 0,730 0,684	2,69 2,56 2,26 2,065 1,89 1,78 1,695	[2,19][6] 1,87 1,855 1,845 1,895 1,864 1,850 Mean: 1,863	[+17,5] +0,4 -0,4 -1,0 +1,7 0,05 -0,7	k1=0,260 ε = $\pm$ 1,0 perc.

Table III
z cm	y cm	β	ψ(β)	k2	δ perc.
0,35 0,45 0,50 0,60 0,70	0,0455 0,0651 0,0760 0,1000 0,1230	0,851 0,766 0,727 0,6615 0,6075	2,147 1,86 1,78 1,66 1,595	1,721 1,736 1,725 1,727 1,655[7] Mean: 1,723	-0,1 +0,7 +0,1 +0,2 -3,9[7]	k1=0,258 ε = $\pm$ 1,2 perc.

Table IV
z cm	y cm	β	ψ(β)	k2	δ perc.
0,150 0,175 0,200 0,225 0,250 0,275 0,300 0,325 0,350 0,375	0,0607 0,0720 0,0835 0,0991 0,1132 0,1290 0,1455 0,1630 0,1813 0,1988	0,963 0,949 0,933 0,883 0,860 0,830 0,801 0,777 0,752 0,732	3,23 2,86 2,73 2,31 2,195 2,06 1,96 1,89 1,83 1,785	8,12 7,99 (?) 7,46 8,32 8,09 8,13 8,13 8,04 8,02 7,97 Mean: 8,09	+0,4 -1,2 [-7,8] +2,8 +0 +0,5 +0,5 -0,6 -0,9 -1,5	k1=0,905 ε = $\pm$ 1,4 perc.

Le $F$ be the intensity of the electric field, $H$½ the intensity of the magnetic field, $k_{1}$ and $k_{2}$ two constants; then:

4)	$q=c\beta ={\frac {F}{H}}{\frac {\eta }{\zeta }}=k_{1}c{\frac {\zeta }{\eta }},$

5)	$\beta =k_{1}{\frac {\zeta }{\eta }},$

6)	${\frac {\epsilon }{\mu }}={\frac {\zeta ^{2}}{\eta }}{\frac {F}{H^{2}}},$

so that under consideration of 1):

7)	${\frac {\eta }{\zeta ^{2}\psi (\beta )}}={\frac {k_{1}c}{H}}{\frac {3}{4}}{\frac {\mu _{0}}{\epsilon }}=k_{2}$

or

8)	${\frac {\eta }{\zeta ^{2}\psi (k_{1}{\frac {\zeta }{\eta }})}}=k_{2}.$

Equation 8) thus represents the equation of the $(\eta ,\zeta )$-curve, derived from the directly measured $(y,z)$-curve by simple transformation.

Thus the task emerges, to determine constant $k_{1}$ by means of the method of least squares, so that the ratio of the left side of 8) is as constant as possible; ${\overline {k_{2}}}$ is the average of all $k_{2}$ found, thus

9)	$\Sigma \delta ^{2}=\Sigma (k_{2}-{\overline {k_{2}}})^{2},$

must be brought to a minimum. Due to the complicated form of $\psi \left(k_{1}{\tfrac {\zeta }{\eta }}\right)$, this can only happen by trying; after some routine one soon will find suitable values of $k_{1}$, namely up to a precision of ½%.

At the end, I report some results of measurement:

Table I is related to my older observations, in which a calculation-error that unfortunately happened at that time and about which I was alerted by E. Gehrcke, was removed. Table II, III, and IV contain the new observations of considerable greater precision,^[8] that were recently made by me with kind support of Mr. and Mrs. Curie, who gave a small quantity of there extremely valuable, pure radium-chloride at may disposal. The enormous activity of this preparation allowed the application of very small granules as radiation source, and a correspondingly fine diaphragm, so that the curves became considerable finer than before, and even the mere voltage of the high voltage battery (ca. 2000 volt) was sufficient, to cause a sufficient separation of the two arms. Curves II and III are recorded with a voltage of 2000 volt, and at curve IV the voltage was increased to ca. 5000 volt by the rotating switch as described l.c. The agreement with theory is so good, as it can be expected by the precision of observation, since the mean error of the individual values only amounts 1 to 1,4% with respect to all four curves.

If the absolute value of $H$ is known, one can also determine $\epsilon /\mu _{0}$ by equation 7. I haven't measured $H$ in the new experiments, while in the old measurements (tab. I) it was $H=299$, from which it is given

10)	$\epsilon /\mu _{0}=1,84\cdot 10^{7}.\,$

in good agreement with the value found for cathode rays:

11)	$\epsilon /\mu =1,865\cdot 10^{7}.\,$

If one calculates (for the experiments in tab. I) the constants $k_{1}$ and $k_{2}$ from the apparatus dimensions, one finds for $k_{1}$ a value deviating by ca. 7,2%^[9], i.e., one doesn't obtain the speed of light for the velocity of the fastest rays, but $2,785\cdot 10^{10}$.

It is very probable, that this differences will vanish with sufficient refinement of the measurement. Experiments in this direction are under way.

Summarizing, it can be said already now, that the observation allow of the following conclusions:

The mass of the electrons forming the Becquerel rays, depends on the velocity; the dependence is precisely representable by Abraham's formula. Consequently, the mass of the electrons is purely electromagnetic in nature.

The value calculated for small velocities, agrees with the value found for cathode rays within the margin of observational errors.

(Self-lecture of the reader.)

Discusson.

Meyer (Königsberg): May I ask, how is the function $\psi (\beta )$ calculated, theoretically or by measurements?

Kaufmann: Perhaps it is better, when we carry out the discussion after the lecture of Abraham.

Abraham (Göttingen): The theoretical derivation will just be given by me; but we can speak about, to what extent the form of function $\psi (\beta )$ (as required by the theory) is confirmed by the observations.

Kaufmann: The comparison with the theory is in the first place made on the basis of the deflections measured on the plate, by determining the two constants depending on the apparatus dimensions and field strengths, not by absolute measurement, but empirically according to the method of least squares.

When one specifies in an absolute way, then it is much harder to reach agreement, because an error of 1% in the determination of $\beta$, already gives an error of 10 or 20% for $\psi (\beta )$. Therefore it is necessary, to compare the relative values with each other. For the time being, an absolute measurement was carried out by me only for the first, older experiments of the previous year. There, one gets deviations of up to 7% for the value of $k_{1}$. If one calculates (after correction of this deviation) from that the value of $\epsilon /\mu _{0}$, then one gets the value $1\cdot 84\cdot 10^{7}$, while $1\cdot 865\cdot 10^{7}$ was found for cathode rays.

Meyer: I want to ask after the shortcomings of the photographic plate: can we recognize them by the fact, that the errors always occur at the same place and the same magnitude?

Kaufmann: These errors are almost always present, however, they are not systematic, i.e., they do not belong to the curve, since they occur upon the single plates at quite different places and they are different with respect to magnitude and sense.

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1. Verhdl. D. Naturf. u. Ärzte Hamburg 1901. II. 1. 45. Gött. Nachr. 1901. H. 2.
2. Phil. Mag (5) 44. 340, 1897.
3. Gött. Nachr. 1902. H. 1.
4. If $a$ is the radius of the electron, then $\mu _{0}={\tfrac {2}{3}}{\tfrac {\epsilon ^{2}}{a}}b$ under the assumption of surface charge.
5. For details of the calculation, see W. Kaufmann, Gött. Nachr. 1902. H. 5. (Although the calculation is carried out in a somewhat different way.)
6. Not used for the determination of the average, since it was obviously distorted by plate errors or other disturbances.
7. Due to the larger inccuracy of the individual adjustment, it was introduced with ¼ weight when calculating the average; the point lies at the outermost viewable end of the curve.
8. The plates will be shown here.
9. W. Kaufmann, Gött. Nachr. 1902. H. 5.