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By S. Fred Singer

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E. that whose dissipation of energy is to be considered. It is clear that, in the transmission of kinetic energy into the medium as pictured in Fig. 1, the impulse received by a negatively charged particle m 1 does not prevent a nearby positively charged particle m2 to acquire an appropriate impulse in the opposite direction, although the charges of m 1 and m 2 may almost completely screen each other in their action upon M. A paradox seems to arise: if m 1 and m 2 are mutually screening, their combined action on M should vanish and M would not lose energy, at the same time imparting energy to m 1 and m2' The paradox is only apparent, and an exact solution of the equation of motion leads, of course, to the PARTICLE DISTRIBUTION IN A FIELD OF FORCE 55 restoration ofthe principle ofconservation ofenergy.

Dv x dv y dvz = dV . dw (c) and the phase-space density D = dN/do. (d) The ordinary space density is p = AN/dV (e) Hence, substituting (a), (c) and (e) into (d), we obtain D = p • I(v) (f) The frequency I (v) is here assumed to depend only on v, the radius vector of velocity space, a simplification which includes the Maxwellian distribution. 56 E. 1. i)PIK In the absence of collisions the velocity of a particle is governed by the field work function e/>, according to the law of conservation of energy tmv 2 = e/> + const which yields, for motion along the dynamical trajectory of the particle, ov/oe/> = l/mv (g) Liouville's theorem requires that, in equilibrium conditions and in the absence of collisions, the phase-space density remains constant along a dynamical trajectory, thus oD/oe/> = 0 (h) Differentiation of (f) by the work function, with substitution of (g), leads thus to the differential equation f(v) .

R (1 2 Z = 90° 2 (217) Zo ~ z ~ 90" ° (218) Q = 1 + cos Zo (219) The orbits escaping at B must have whence, integrating (108) from z = to z = ,"/2 for the inward flux and from z = Zo to z = ,"/2 for the outward flux, we obtain FIG. 6. to be used in (109) at constant U. /m_us~2)J 1{2 R Q = 1 + [ 1 - r2(1 + 3tP,/mu;) (220) with mu; = 3kT for thermal motions. The same formula can be used as an approximation in (124) for repulsion, tP < 0, with the understanding that when 1 + tPR/kT < 0, Q = 2 is to be taken.

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