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§12. – ELECTRIC OSCILLATIONS. GENERAL CONSIDERATIONS.
Translated from Recherches
critiques sur l'Ėlectrodynamique Générale,
Annales de Chimie et de Physique, Vol. 13, p. 145, 1908.
We have seen in the preceding sections, that for very rapid
oscillations, only the accelerations are
to be retained
besides the electrostatic term, the (effect of the) speeds being
negligible compared to them. It will no longer be permitted to refer to
developments which we have used up until now, and we must go back to formula
(VI), using u=0. It then becomes
In Lorentz’s theory, only the electric force will play a role
(Oeuvres 404) and we can
also neglect the terms in so that it (the expression) becomes
and this is for
any value of R which would not be incomparably smaller than 1 cm, and a
fortiori (even more so) for large values.
The second terms of the two formulas (31), (33) correspond to a force
perpendicular to the radiation vector,
which in optics
(where the term in 1/r2 disappears) plays the role of Fresnel’s
vector. It depends only on x, y, z, t, and not on v; the
notion of field may thus apply, that is to say that we may consider the
distribution of this force in space independently of the presence or absence of
an electron, which would no longer be the case if it depended on the speed of the
General discussion. – Formulas (VI) and (31) demand a
general remark. They contain, in effect, the factor, which becomes infinite for sufficiently large values of r.
I am going to show that there nevertheless result from these expressions only
Let us consider a charge e' vibrating
parallel to the x-axis and whose speed v'= v'(t') is nil for t'=0 and
afterwards positive. The wave
the length of the axis with the speed c, and reaches a fixed point P
situated on the positive axis at an instant t>t'>0. Then
the speed of propagation increases, and will be c+v' for the
wave emitted at . This wave will therefore overtake the previous one
at a certain distance x0, then will pass it. At point x0 the two
waves emanating from e' at t'=0 and will arrive
simultaneously. The speed v' remaining positive, the number of
waves arriving simultaneously at x0 may still increase; but,
diminishes or changes sign, this case no longer presents
itself. Where the position of P and the movement of e' are
given, the time of arrival t is a function f(t') clearly
determined from t', given by the law of propagation [analytically by
formula (IIa) (Oeuvres 405) resolved in relation to t]. On the contrary, the function t'=φ(t) may present, for a given value of t, an finite number of distinct determinations as we have just seen. Let us show t'
t in abscissas, and suppose the relative speeds are greatly
inferior to c; the curve t'=φ(t) will
oscillate from side to side of the straight line , where r0 is the average distance from e' to P.
Parallels to the t-axis will not cross it except at one point; parallels to the
t'-axis, at one of several points; among these latter, the tangents to the
curve correspond to the points where becomes infinite and where the two values become
equal. If remains finite at
this point (that is to say if there is no inflexion), the values of will tend towards great values and of opposite signs.
These considerations are clearly general and apply even when P is
in movement, as soon as the relative speeds become less than c.
We have seen [formula (6) obtained by differentiation of (II)] that
The denominator therefore annuls itself and changes sign precisely
at the instant when two different values are corresponding to a same
instant of reception, and tend to become equal. But it has been specified (§2) that
the action of e' on e is the sum of the
actions corresponding to Let us multiply and divide Fx (VI) by . This expression will be written
where Φ remains finite and continuous for
the instant where and become infinite. The sum of the terms
relative to and
(Oeuvres 406) will remain finite2,
because and for the critical instant.
is unlimited in , this reasoning is not applicable any more; but these cases
are exceptional, and we have to keep account here of the finite dimension of
the electron: the impetus because it remains finite in all these cases.
Translation editor notes:
[*] In expression (32) there needs to be a c^2 in the denominator of
the third term inside the first upper parenthesis.
[**] Even though Ritz uses the word propagation, which
would seem to imply a wave action impressed on a deformable medium, his
mathematical reasoning is more in line with the idea of ballistic projection of his
fictitious particles. (In Lorentzian
ether there is only one speed of propagation, i.e., the speed