— 12 —
V.
The advantages arising from the formulation of the world-postulate are illustrated by nothing so strikingly as by giving the expressions for the reactions exerted by a point-charge moving in any manner according to the Maxwell-Lorentz theory. Let us conceive the world-line of such a point-like electron with the charge e, and let us introduce upon it the proper-time reckoned from any initial point. In order to obtain the field caused by the electron at any world-point , let us construct the fore-cone belonging to (vide fig. 4). Evidently this cuts the unlimited world-line of the electron at a single point P, because these directions are all time-like vectors. At P, let us draw the tangent to the world-line, and let us draw from the normal to this tangent. Let r be the sum of . According to the definition of a fore-cone, is to be reckoned as the sum of PQ. Now at the world-point , the vector with respect to PQ of magnitude in its components along the x-, y-, z-axes, is represented by the vector-potential of the field multiplied by c; the component along the t-axis is represented by the scalar-potential of the field excited by e. This is the elementary law found out by A. Liénard, and E. Wiechert.[1]
In the description of the field caused by the electron itself, then it will appear that the division of the field into electric and magnetic forces is a relative one with respect to the time-axis assumed; the two forces considered together can most vividly be described by a certain analogy to the force-screw in mechanics; the analogy is, however, imperfect.
I shall now describe the ponderomotive force which is exerted by a point-charge moving in an arbitrary way, to another point-charge moving in an arbitrary way. Let us suppose
- ↑ A. Liénard, Champ électrique et magnétique produit par une charge concentrie en un point et animée d’un mouvement quelconque, L’Éclairage électrique 16 (1898), p. 5, 53, 106; Wiechert, Elektrodynamische Elementargesetze, Arch. néerl. (2), 5 (1900), p. 549.
