Page:Prinzipien der Dynamik des Elektrons.djvu/1

Principles of the Dynamics of the Electron

M. Abraham (Göttingen)

Already in January this year, I have published a treatise on the dynamics of the electron in the "Nachrichten der Göttinger Gesellschaft der Wissenschaften". The agreement of the theories developed there, with the experimental results of Kaufmann, makes the assumptions on which the theory is based, look appropriate; it furthermore shows, that the inertia of electrons is purely electromagnetic in nature. While I at first still used a "material" mass independent of electric charge, it becomes necessary now, to establish the dynamics of the electron from the outset in an electromagnetic way. From that, remarkable analogies of the principles of the electron on one hand, and the principles of ordinary dynamics of material bodies on the other hand, are given, analogies which may become important for the coming electromagnetic foundation of the whole of mechanics.

We ascribe to the electron, the atom of negative electricity, a charge $e$ (measured electrostatically). The free electron moving in the cathode- and Becquerel rays shall be (as we suppose) a sphere of radius $a$, where electricity is uniformly distributed on its volume with density $\varrho$. Electricity shall be connected to the volume elements of the electron, as matter to the volume elements of a rigid body, i.e. the kinematic fundamental equation shall apply to the electron.

1)	${\mathfrak {v}}={\mathfrak {q}}+[\vartheta {\mathfrak {r}}].$

The kinematic fundamental equation determines the velocity ${\mathfrak {v}}$ of any point of the electron, whose distance from the center is indicated by a vector ${\mathfrak {r}}$, when the velocity ${\mathfrak {q}}$ of the center and the angular velocity $\vartheta$ around the center are given; we write them in vectorial form, and we use Grassmann's symbol of the outer product.

The electromagnetic field agitated by the electron, is determined by the Maxwell-Lorentz field equations:

2)

${\begin{array}{ll}{\frac {1}{c}}\cdot {\frac {\partial {\mathfrak {E}}}{\partial t}}=curl\ {\mathfrak {H}}-{\frac {4\pi \varrho }{c}}\cdot {\mathfrak {v}},&div\ {\mathfrak {E}}=4\pi \varrho ,\\\\-{\frac {1}{c}}\cdot {\frac {\partial {\mathfrak {H}}}{\partial t}}=curl\ {\mathfrak {E}},&div\ {\mathfrak {H}}=0.\end{array}}$

there, ${\mathfrak {E}},\ {\mathfrak {H}}$ denote the electric and magnetic field strength, $c$ the speed of light.

It is to be emphasized, that Lorentz's theory calculates with absolute velocities.

The electron shall now be located in a given external electromagnetic field, of field strengths ${\mathfrak {E}},\ {\mathfrak {H}}_{h}$. For the determination of the motions carried out by it, also a third system of fundamental equations is required, the system of "kinetic" or "dynamic" fundamental equations. In the course of stating these equations, we will be guided by the following reasoning. H. A. Lorentz and E. Wiechert have shown, that one can derive the forces acting upon resting and streaming electricity in the electric or in the magnetic field, from the electron theory by making the following assumption for the force acting on the individual electron:

${\mathfrak {R}}=e\cdot {\mathfrak {F}}_{h},\ {\mathfrak {F}}_{h}={\mathfrak {E}}_{h}+{\frac {1}{c}}[{\mathfrak {q}}\ {\mathfrak {H}}_{h}].$

There, the electron is interpreted as a point charge. We have to distinguish the volume elements of the electron; therefore we define the "outer force" by

${\mathfrak {R}}=\int \int \int dv\ \varrho \cdot {\mathfrak {F}}_{h},\ {\mathfrak {F}}_{h}={\mathfrak {E}}_{h}+{\frac {1}{c}}[{\mathfrak {v}}\ {\mathfrak {H}}_{h}]$

and furthermore introduce the "outer angular force"

$\Theta =\int \int \int dv\ \varrho [{\mathfrak {r}}\ {\mathfrak {F}}_{h}]$.

According to Maxwell and Hertz, however, the "principle of the unity of electric and magnetic force" applies; according to this principle, the separation of an "outer field" and an "inner" field excited by the electron itself, is artificial; in reality only one field exists, of field strengths:

${\mathfrak {E}}_{h}+{\mathfrak {E}},\ {\mathfrak {H}}_{h}+{\mathfrak {H}}.$

This principle leads us, so as to put the vector

${\mathfrak {F}}_{h}={\mathfrak {E}}_{h}+{\frac {1}{c}}[{\mathfrak {v}}\ {\mathfrak {H}}_{h}]$

next to the vector

${\mathfrak {F}}={\mathfrak {E}}+{\frac {1}{c}}[{\mathfrak {v}}\ {\mathfrak {H}}]$,

and to speak about an "inner force"

$\int \int \int dv\ \varrho {\mathfrak {F}}$