# Translation:On the Apparent Mass of the Ions

On the Apparent Mass of the Ions  (1900)
by Hendrik Lorentz, translated from German by Wikisource
In German: Über die scheinbare Masse der Ionen, Physikalische Zeitschrift. 2, 1900/1, pp. 78-80

H. A. Lorentz (Leiden)
On the apparent mass of the ions.

It is known that by observations of cathode rays we were able to derive the ratio ${\displaystyle {\tfrac {e}{m}}}$, i.e. the ratio between the charge of an ion ${\displaystyle e}$ and its mass ${\displaystyle m}$. The question arises, what is meant by that mass. In any case we must attribute an apparent mass to the ion, as it generates a certain energy in the ether by virtue of its motion. This apparent mass will be denoted by ${\displaystyle m_{0}}$. It is possible that the ion also possesses a real mass in the ordinary sense of the word; in this case, ${\displaystyle m_{0}. If this is not the case, then ${\displaystyle m_{0}=m}$.

So we have the inequality

${\displaystyle {\frac {e}{m_{0}}}>{\frac {e}{m}}{,}}$

when there still is a real mass besides the apparent mass; otherwise

${\displaystyle {\frac {e}{m_{0}}}={\frac {e}{m}}.}$

So we want to write

${\displaystyle {\frac {e}{m_{0}}}\geqq {\frac {e}{m}}{,}}$

where ${\displaystyle {\tfrac {e}{m}}=10^{7}}$ is.

Now

${\displaystyle m_{0}={\frac {8}{3}}\pi R\sigma e{,}}$

if we conceive the ion as a sphere, ${\displaystyle R}$ is the radius of this sphere, and ${\displaystyle \sigma }$ means the surface density of the charge.

This formula allows for an interesting conclusion on the radius of the ions. If, namely, we substitute for ${\displaystyle m_{0}}$ the now specified value into the inequality, we obtain an inequality for the radius. We have

${\displaystyle 4\pi R^{2}\cdot \sigma =e{,}}$

thus

${\displaystyle m_{0}={\frac {8}{3}}\pi R\sigma e={\frac {8}{3}}\pi Re\cdot {\frac {e}{4\pi R^{2}}}={\frac {2e^{2}}{3R}}}$

and thus

${\displaystyle {\frac {e}{m_{0}}}={\frac {3R}{2e}}{,}}$

and

${\displaystyle {\frac {3R}{2e}}\geq 10^{7}}$

and

${\displaystyle R>10^{7}\cdot {\frac {2}{3}}e.}$

The magnitude ${\displaystyle e}$ is unfortunately not known. If we take the charge of an ion in a cathode ray to be as great as in an electrolytic hydrogen, and presuppose the size of a hydrogen molecule, we obtain for ${\displaystyle R}$ a magnitude of order ${\displaystyle 10^{-12}}$ cm, that is certainly not an arbitrarily small magnitude, but a lower limit.

The question of whether or not a real mass exists besides the apparent mass of an ion, is extremely important; because by that we touch the question of the relation of ponderable matter with ether and electricity. I am far away to announce a decision, but I would like to cite but a few questions whose resolution can potentially bring us further in that question.

The first question is whether an ion rotates in a magnetic field. Actually, we should expect that. Since if an ion is present, and if a magnetic field is caused, then a rotation arises, as it can easily be derived from the formation of induced currents. Of course this is also the case when the ion flies into an already existing magnetic field. The velocity of rotation will depend on the magnitude of the mass; if only apparent mass is present, and even a corresponding moment of inertia, then the rotation velocity has a certain value. If, however, a real moment of inertia is added, the rotation is slowing down. Unfortunately I can not find any phenomenon, from which we could conclude anything about this rotation.

A second means by which we maybe could decide the question of the relationship between the apparent and real mass is the following:

The value for the apparent mass was given above only in first approximation. If the velocity is such that it is comparable to the velocity of light, then additional magnitudes will be added. For a straight path of the ion we can calculate the intensity of the field and the size of the energy and deduce from that the mass factor. In general, the trajectory will be curvilinear through the influence of the magnetic field, e.g. circular; then the calculation of the mass factor will become more complicated, but it can be carried out. If we denote by ${\displaystyle m_{0}}$ the expression above and ${\displaystyle q}$ is defined as the ratio of the ion velocity to that of light, it follows in second approximation for the apparent mass of the ion in linear motion:

${\displaystyle m_{0}\left(1+{\frac {6}{5}}q^{2}\right){,}}$

while in a circular motion the term with ${\displaystyle q^{2}}$ yields a different coefficient.

These terms of the second order could now perhaps become observable, because the velocity of cathode rays increases up to a third of that of light, hence ${\displaystyle q={\tfrac {1}{3}}}$ and ${\displaystyle q^{2}={\tfrac {1}{9}}}$. To come to a decision, we could think of experiments as they were done by Lenard, to examine the influence of electric forces on the velocity of cathode rays. He has shown that the magnetic deflectability of the cathode rays, which is of course the smaller, the greater the speed, will change when the rays can pass through the space between two charged capacitor plates in the direction of the electric force lines.

We could measure the magnetic deflection in the case of an uncharged capacitor, then in the case of charge in one direction and then for the other direction. Thus we would obtain three different values of deflectability, between which a simple relation should exist, if the terms of second order could be neglected. If we measure each time the magnetic field-force required for a particular deflection, then the squares of these three field forces should form an arithmetic row. A deviation from this relationship would indicate that the terms with ${\displaystyle q^{2}}$ shall not be neglected, and that therefore in any case the apparent mass is noticeable. Detailed specifications could decide concerning the ratio between the real and the apparent mass, and concerning the question whether a real mass exists. It turns out that by Lenard's experiments we were near to decide about the existence of terms of the second order.

(Self-lecture of the lecturer.)

Discussion. (Reviewed by the participants.)

W. Wien. I was recently concerned with similar issues, and would like to stress that Lenard has observed cathode rays at low velocities, triggered under the influence of ultraviolet light. There, he found a small value for the ratio of mass to charge, namely the decrease lies in the sense which is required by the theory.

I have tried to transcend over Lorentz's position, by posing me the question, whether it would suffice when we only consider the apparent mass and omit the inertial mass, and replace it with the electromagnetically defined apparent mass to present the mechanical and electromagnetic phenomena in an uniform way. Because the magnetic and mechanical phenomena are only connected by the energy principle so far. I've tried to pose the question as to whether we could try by Maxwell's theory, to involve mechanics as well. The possibility of an electromagnetic explanation of mechanics was given, after Lorentz has developed a conception of the law of gravity, according to which it would be very similar to electrostatic forces. We would have to think of matter as only composed of very small positive and negative charges, which are within a certain distance from each other. By this condition, the ponderable mass is not constant but depends on the velocity, and namely we obtain terms, depending on even powers of the ratio of velocity to the velocity of light. The numerical factor by which the second term is multiplied, depends on the curvature of the trajectory, but also on the shape of the electric charge. Depending on which different way we choose the form of electrified molecules, we come to other numerical factors. Concerning the ordinary motions on earth, it vanishes because the velocity is very small. Concerning planetary motions we probably can achieve something; because we reach velocities at which we have to consider the terms of second order. On the assumption of a specific type of charge, leading to the simplest electromagnetic field, these terms become relevant in a way, so that the accelerations of two bodies by gravitation are the same up to a slightly different numerical factor, as if the bodies attract each other with constant mass according to Weber's laws. The electromagnetically defined mass comes into play, as if not Newton's, but Weber's law would apply.

Lorentz. In essence, we agree; but Wien already wants to go further than I do. Anyway, it seemed of interest to me to look for means, by which we can come to a decision on the issue discussed. One more thing I would like to add: I made the assumption that the sphere, which forms an ion, is rigid. But perhaps one might think that the sphere would be transformed into an ellipsoid when in motion. This has some similarity with the diversity, that was pointed out by Wien.

Voigt. I would like to pose the question to the lecturer, concerning the reflection of cathode rays; should a rotating ion not be reflected differently, as a non-rotating one?

Lorentz. Certainly, if one imagines that the reflection happens on a surface. But if you look at the reflection, which is more likely to me, as caused by forces that occur at some distance from the surface of the ion, then those surely act on the center, and then the influence of rotation vanishes.

Warburg. What does the theory say about the velocity of the ions during reflection? Does it remain the same?

Lorentz. As far as I know, yes. I have not elaborated on this.

Warburg. Merritt has found that the velocity of reflection has not changed. But the experiments of Cady on the energy of cathode rays are in contradiction to this, so I've thought that the experiments of Merritt may not be completely correct, and maybe we could obtain a velocity change. I wanted to ask if the theory says something in this respect.

Lorentz. I can not say this right now.