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### INDUCTION

#### §9. –THE __INDUCTIVE__ CURRENT IS NEUTRAL
AND CLOSED

WALTER RITZ

##### Translated from Recherches
critiques sur l'Ėlectrodynamique Générale,

*Annales de Chimie et de Physique,* Vol. 13, p. 145, 1908.

Let *ds**'** be a linear element of one of the closed currents in which
one can decompose the inductive current.* Let *V**'** be the speed of the element (which is at the same time that
of the positive ions); we *(Oeuvres 398)** ***have
calculated (27) the action of such an element on a mobile ion by supposing **V**'=**0 and neglecting the
accelerations. Let us go back to the calculations without these restrictions,
but also ignoring the electrostatic action of the current, and, **a fortiori**, all the
terms multiplied by the resultant charge **E**'** of **ds*, which
comes back to setting _{}. Let *F* be the force exerted** **by the positive
and negative charges of *ds* on a unit** **of charge placed in (*xyz*),
of speed *v*. The electromotive force of induction produced in a conductor
wire whose element is *ds* (*dx,dy,dz*) is determined by the total
tangential component of *F*:

_{}

In closed
induced currents, we observe only the value of this integral taken along a
closed contour. If therefore *F* contains terms of the form _{}, these terms will give a total differential, and
consequently will disappear from the result.

To calculate *F*, let us start from formula (15). We can neglect the
term independent of *v*,_{}for all closed induced currents. The term containing *w*
will be multiplied by *E**'**, and in consequence *__will__
obviously __be__ nil; thus we will have

Annales 243

_{}

In this expression, the terms independent of *v* will not figure in
the __operation__ _{}, and their partial derivative in relation to x y z alone
intervenes and gives, from that point, a nil result. There will remain

_{}

(Oeuvres 399) The first term

_{}

gives, integrated
by parts,

_{}

so that we have
in the end

_{}

In Lorentz’s theory, there exists a** **function *L* given by
(20a) and which leads, by the same operation _{}to the expression of the force. The terms in _{}of this expression again give zero

Annales 244

for a closed __secondary__
circuit. The resulting value of *L* will thus be

_{}

*Thus the two theories give, for two closed circuits in motion*
(for we have made no hypothesis on the movement of the induced circuit, and we
have not supposed *v* parallel to *ds*), *the same result*, in
accord with the experiment.

The more special hypotheses of Weber led,** **as we know, to the same
results.

Magnets will be treated as in Lorentz’s theory.

One remark is nevertheless necessary. In order for Lorentz’s formula to conform
with

experiment, it is
necessary that only the tangential acceleration _{}play a role (in the case of variable currents), the normal
acceleration, proportional to

being negligible
beside it. Stated otherwise necessary that

_{}

must** **be
essentially reduced to its first term.

(Oeuvres
400) When *the *__secondar__y circuit
is open, there occur, as we know, potential differences; the study of
several cases where these differences could be observed may easily be treated
by the same methods used up to this point, and I won’t dwell on it.