Quantitative Theory of the Influence Electrostatic Generator

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PHYSICS: A. W. SIMON

PRoc.. N. A. S.

It should be noted that in the curve showing the higher orders, the ordinates do not begin at zero ionization current, so that the peaks are really on a rather high base level made up primarily of scattered radiation and higher o'rders of the reflected general -radiation. This. would nmake them very difficult to detect by a photographic method. Recently Walter5 has attempted to obtain photographic evidence of this phenomenon from crystals of KBr with m,olybdenum primary rays as in these experiments,. and has reported. negative results.. His photographed spectra cover the first and second orders of the molybdenum K series, and the first order of the bromine 'absorption only. From the curve presented here,it would seem almost impossible to obtain photographic evidence for the presence of Bra in the first order due to its diffuseness and the low intensity in this region. The evidence for the presence of this phenomenon in all cases comes primarily from higher orders. Other experiments will be reported by Clark and Duane which indicate that in order to obtain photographic evidence of the presence of these peaks, a narrow slit in front of the photographic plate should be used to limit the reflected beam. Such a slit is not usually employed in photographing X-ray spectra. 1 NATIONAL RusEARCH FELLOW. These PROCESDINGS, 9, P. 126; 10, p. 419 (1923). 3 Wyckoff, J. Wash. Acad. Sci., 11, 429 (1921). 4 Baxter and Wallace, J. Amer. Chem. Soc., 38, 259 (1916). 5 Zs. Physik, 20, 257 (1923). 2

QUANTITATIVE THEORY OF THE INFLUENCE ELECTROSTA TIC GENERA TOR By ALPRD W. SIMON

RYZRSON PHYsIcAL LABORATORY, UNIVERSITY

OF CHICAGO

Communicated June 4, 1924

A rigorous quantitative theory of the influence electrostatic generator has never been given. Maxwell' has worked out a special case-that of the Kelvin replenisher-but his solution is only an approximation, for he has assumed that the carriers give up all their charges to the inductors, and, further, has neglected the fact that each element of the machine is influenced by all the other elements. The author has developed a mathematical treatment, which can be applied to any static machine, and which is not subject to the criticisms mentioned above.

VOL. lo, 1924'

PHYSICS: A. W. SIMON

303

The method of attack consists in (1) writing the equations for the charges on the elements of the machine in terms of the potentials, the coefficients of capacity, and the coefficients of induction of the elements, for each of those configurations of a complete cycle in which a change of charge occurs on any of the elements. To these equations are added a number of others involving the charges only, and usually expressing conservation of quantity of electricity. Leakage is neglected. (2) From this basic group of equations all the charges are eliminated, and a number of simultaneous equations obtained, each of which gives the potential of a particular element at, say, the beginning of the (n + 1)st cycle in terms of the potentials of a certain number of the elements at the- b-eginning of the previous, the nth, cycle. (3) From this second group of equations there is deduced a number of additional equations giving the potentials of' the same elements at the beginning of the (n + 2)nd, (n + 3)d, up to the (n + m)th cycle. (4) From the latter set, by elimination, a final set of equations is deduced, each of which involves only the potentials of a single element over a number of successive cycles, and which is'therefore of the form: (1) Vn+m + kiVn+m-l + .... + km Vn = 0 where kl, k2, etc., are certain combinations of the coefficients of capacity and coefficients of induction involved in the basic set of equations, and VIn + m is the potential of a certain element at the be'ginning of the (n + m)th cycle. Each one of these equations expresses the potential of a given element in the form of a recurrent sequence, which can be evaluated by the calculus of finite differences. The general solution of (1) is:2

Vn = Clrl + C2r2 + ..+ Cmrm where ri, r2 ... rm, are roots of the algebraic equation

(2)

rm + kirml +.... kmrm = ° The roots ri, r2, r3, etc., can usually be'determined by inspection from the coefficients ki, k2, k3, etc., and are functions only of the coefficients of capacity and induction involved in the original equations. The constants Ci, C2, C3, etc., depend on both the original values of the potentials of the elements of the machine, and the geometry of the machine (coefficients of capacity and induction). In general, not all the roots are greater than-unity so that as n increases a certain number of terms appearing in (2) approach zero as a limit. Some of the roots may be negative, and the corresponding terms represent oscillating sequences, giving the theory of the alternating electrostatic generator.3 The theory has been applied in detail to a four-element static machine

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.PROC. N. A. S.

of the type exemplified by the multiplier of Belli, the machine of Varley, or the replenisher of Lord Kelvin.' For this case it is found that the potential of one of the inductors is given by an equation of the form:

C1r'

+ C2r V. = where, in general, ri is less than unity and r2 greater than unity.

1 J. C. Maxwell, A Treatise on Electricity and Magnetism, 1, p. 110, 3d. edition, 1892. G. Boole, A Treatise on the Calculus of Finite Differences, Chap. 11. 3 J. Wimshurst, Phil. Mag., 31, p. 507, 1891. 2

SCA TTERED X-RA YS By P. A. Ross STANFORD UNIVURSITY Communicated May 12, 1924

In a previous issue of these PROCBSDINGS' the writer published a photograph of the molybdenum K lines scattered at 90 from a paraffin block. This spectrogram showed an unshifted scattered line and a shifted line agreeing in position with that required by Compton's quantum theory of scattering.2 It has since been suggested3 that the shifted line shown in the spectrogram might be explained by the tertiary radiation theory of Clark and Duane.4 More recent and much better spectrograms make it clear that this explanation is not tenable. From the spectrograms here published which were shown at the Pasadena Meeting of the American Physical Society in September 1923,5 it may be seen that there is a shift of the magnitude required by the Compton theory not only when the radiation is scattered by light elements but also an exactly equal shift with a heavy element such as lead. It will be seen that as we proceed toward the heavy elements the intensity of the shifted as compared with the unshifted line falls off very greatly. Spectrograms number 6 and 7 show the variation of the intensity and amount of shift with angle of scattering. The amount of shift at 30° and 600 agrees with that required by theory. That this line cannot be due to a strongly filtered tertiary peak from carbon and the shift and decrease of intensity due to greater filtering at small angles is proved by the fact that at 300 the modified line has moved past the short wave-length limit of the tertiary peak computed from the carbon K absorption limit. In photographs 1 to 7 inclusive a water cooled molybdenum tube was used at an energy input of about 30 to 40 ma. at 30 kv. for 90 to 140 hours.