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The editors cannot undertake to do more than correct obvious minor errors. REPRINTS should be ordered at the time of submission of manuscript. They will be furnished to authors at cost, approximately as follows: (The charge for fewer than 100 copies is practically the same as for 100.) Copyright, 1919, by the National Academy of Sciences Communicated by R. A. Millikan, June 7, 1919 1. The rate of radiation of energy from a ring of electrons revolving in a circular orbit and from various other distributions of moving electric charges and magnetic poles has been calculated by G. A. Schott,1 who finds that the rate of radiation of energy is almost invariably positive. This is certainly true in the case of a single electric pole describing a circular orbit as is indicated by the well known formulae of Larmor and Liénard for the rate of radiation. Thus electromagnetic theory in its present form lends no support to Bohr's idea of non-radiating orbits. A steady distribution such as a Parson magneton which consists of a complete ring of electric charges following one another round the ring at a constant speed will evidently give no radiation when the ring is stationary as a whole, but as Schott remarks the ring may be expected to radiate energy when its centre has an acceleration. Schott's results are so important that it is desirable that they should be confirmed by an independent method and an attempt has been made to devise a method by which the rate of radiation from a moving electric pole and magnetic doublet may be readily calculated. In two important cases we have confirmed Schott's surmise that the rate of radiation is positive. 2. Starting with the case of an electromagnetic doublet, i.e. an electric doublet and magnetic doublet which move together, we determine the electric force E and the magnetic force H from the equations = §' (7) [x − § (7)] + n' (t) [y−n (7)] + 5′ (7) [≈ − 5 (7) ] — c2 (t−7) V = = r [(v·s) — c] y2 = [x− §(7)]2 + [y − n (7)]2 + [% − 5 (7)]2 = c2 (t − 7)2. T tZT. In these equations v denotes the velocity at time of the moving point P whose coordinates at this instant are (7), N(T), 5(T); x, y, z are the coordinates of an arbitrary point Q; s is a unit vector in the direction of the line PQ; p and q are vectors representing the electric and magnetic moments at time 7; t is the time, and c the velocity of light. From these equations we find that E× H = c2 — s[g* · g* — (s•g*) (s · g*) where g* is a certain complex vector and g*. the conjugate complex vector. A similar expression may be obtained in the case of an electromagnetic pole and also in the case of the combination of an electromagnetic pole and an electromagnetic doublet. In the latter case the appropriate expression for g* is Here 4πе and 4h are the electric and magnetic charges associated with the pole and primes denote differentiations with respect to 7. Calculating the rate of radiation I across a very large sphere whose centre is at the moving point P we find that I may be represented by the real part of the following expression 1 +3v′2 (g′ · gó) + 4 (v · v′′) (b · gɔ) + 2 v′2 (b · go) + 2 (v′ · v′′) (g'·go)+ · (8 · 80) } 64 2c2 + v2 24 (v · v')3 (g′ · go) + 9 v′2 (v · v′)2 (g ⋅ go) 7 (c2-02)7 (c2 3 8 1 5 (c2 (c2 — v2) 4 6 (v · b) (v' · g') + 6 (v · gó) (v' · b) (v′ · v′′) (g′ · go) + (v′′ · g′) (v′ · 80) (2′′ · go) + 2 12 (v · v′) (v · gó) (v · b)+6(v·v') (v·g′) (v' · gó) +6(v· v′) (v'• g′) (v• gó) 5 (c2-2)5 +3v'2 (v·g') (v·gó) + 2v′2 (v · b) (v · go) +4(v·v′) (v·b) (v′ · go) + 4 (v · v′) (v′ · b) (v · go) + 2 (v' · g′′) (v · g') (v · go) + 2 (v · v') (v · g′) (v′′ · go) + 2 (v · v′′) (v · go) (v′ · g′) 0") · · · · + 2 (v · v′′) (v · g') (v′ · go) + 2 (v · v') (v · go) (v′′ · g′) + = (v.v′′) (v · g) (v′′ · go) 3 4 +Hz(0.8) ( v′′2 (v · g) (v · go) + —— v′2 (v · g′) (v′·go) + —v'2 (v · go) (v′· g′)+7—7 (v·v′′) (v′· g) (v′· 80) (0%) + — · · ; · 4 36 += (v·v') (v' · go) (v" · g) + 4(v⋅ v") (v · b) (v⋅ go) + 7 (v · v') (v' · g′) (v′ · go) 7 + 2 (v ·v′)2 (v · b) (v · 80) + 2 (v·v′) (v·0′′) (v·g′) (v·go) + ¦ ¦ (v·0′′)2 (v · 8) (v ·80) 1 64 3(v·v')2(v⋅g') (v⋅gó) 1 (v·v′)2(v′·g) (v′·go) + — v′2 (v · v′) (v· g) (v′· go) + 24 (v · v′)3 (v · g′) (v · go) + 8(v · v′)2 (v · v′′) (v⋅ g) (v · go) + 9v′2 (v · v′)2 (v · g) (v· go) (v · v′)1 (v · g) (v · go) — iπc2 4 7c2 + v2 {4 (v · v′′) [v · (b × go)] 5 (c2 — 2)5 + 6 [v' · (b × g')] + 2 [v” · (b × go)]} + + 12 (v · v') [v · (b × gó)] + 6 (v· v′) [v′ · (g′ × g')] + 3v′2 [v · (g′ × gó)] + 4 (v · v') [v' · (b × go)] + 2v'2 [v⋅ (b × go)] + 2 (v.v′) [v" · (g' × go)] + 2 (v · v′′) [v′ · (g′ × go)] + 2 (v′ · v′′) [v · (g′ × go)] 2 + = (v · v′′) [v′′ ·(g × go)] 3 a'2 [a′ · (8′ × 80)] + — (a′ · a′′) [d′ · (8 × 80)] [v' 16 9c2+v2 5 (c2-2)6 2 7 · { 6 (v · v')2 [v · (g'×g')] +4 (v · v′)2 [v· (b×go)] . + 4 (v · v′) (v· v′′) [v · (g′× go)] + = (v · v′′)2 [v · (g×go)] + = v'2(v · v′) [v′· (g×go)] { 24 (v · v')3 [v · (g′ × go)] + 12 (v · v′)2 (v · v′′) [v · (g × go)] 3. In the special case of a magneton of charge 4re and moment 4rk describing a circular orbit at a constant speed and in such a way that the axis of the magneton is always perpendicular to the plane of the orbit, we find that if k is constant the rate of radiation of energy is 8 3 8 If e, v and v' are given the minimum value of this positive quantity is found to be 2π ce2y/2 c2+4v2 3 (c2-v2)2 2c2+3v2 and is thus about of the rate of radiation from the electric charge alone. Some years ago Dr. W. F. G. Swann expressed to me the desirability of calculating the radiation from an electron which rotates about its axis like a planet while describing a circular orbit. If such an electron can be treated as a magneton to a first approximation the above result is applicable. The fact that the radiation is reduced by rotation may indicate that revolving electrons do rotate. |