THE AMERICAN JOURNAL OF SCIENCE [FOURTH SERIES.] ART. I.-An Experimental Investigation into the "Skin"effect in Electrical Oscillators; by C. A. CHANT. 1. Introductory and Theoretical. THE first explicit reference to the fact that, when a conductor is subjected to a periodic electromotive force, the current is not uniformly distributed over a cross-section of the conductor, is to be found in Art. 690 of Maxwell's Electricity and Magnetism. Upon obtaining the equation connecting the impressed electromotive force with the effective electromotive force and the inductive electromotive force he introduces terms which "express the correction of this value [of the inductive electromotive force] arising from the fact that the current is not of uniform strength at different distances from the axis of the wire. The actual system of currents has a greater degree of freedom than the hypothetical system in which the current is constrained to be of uniform strength throughout the section. Hence the electromotive force required to produce a rapid change in the strength of the current is somewhat less than it would be on this hypothesis." It is quite certain, however, that Maxwell did not foresee the great interest and importance which the subject was destined soon to develop. In a series of papers written between 1884 and 1887 Heaviside* dealt with the entire question of the propagation of electric currents into conductors and of magnetization into. cores when produced by a periodic electromotive force. He was one of the first to insist that the action should be considered as entering the conductor from the surrounding dielec* Electrical Papers, vol. i, pp. 353, 429; vol. ii, p. 168. AM. JOUR. SCI.-FOURTH SERIES, VOL. XIII, No. 73.-JANUARY, 1902. tric. He compares the transmission of the effect into the metal to the transmission of motion into the inner portions of liquid in a cylindrical vessel when the vessel is given a rotatory vibration about its axis. Especial attention was drawn to the subject by Hughes,* who treated the question experimentally. In 1886 Rayleight published his well-known paper in which he obtained expressions for the resistance and self-induction of a straight conductor carrying a periodic current. For very rapid oscillations the resistance in which is the length of the conductor, μ its magnetic permeability, R its resistance to steady currents and p = 2πn, where n is the frequency. In 1890 Stefan, in a paper on electric oscillations in straight conductors, also obtained formulas for the resistance and selfinduction. With very high frequency his expression for the resistance is ance. where a is the radius of the conductor and σ its specific resistThis formula is equivalent to that given by Rayleigh. He remarks that for very great frequencies metallic conductors act much as though without resistance, but electrolytes behave very differently on account of their very high resistance. He finds that for a cylindrical copper conductor 1cm in diameter, with a frequency of 50 millions, the current density at a depth of 0-004cm is only 1/100 of that at the surface; while for a tube of equal size of carbon disulphide the current density at the center is but 0.8 per cent lower than at the surface,--in other words, the current is practically uniform. If, now, the action enters the conductor from the surrounding dielectric and is prevented from penetrating very far by the rapidity of the oscillations, it is evident that very thin layers of metal should be sufficient to ward off electrical undulations, either by absorption or reflection. In a paper published in 1889 Hertz§ described experiments made to find out how thick a metallic film was needed to screen from his rapid oscillations. Tinfoil, Dutch metal and gilt paper acted perfectly. The thickness of the metal on the latter he estimated at 1/20mm though it was probably much *Jour. Soc. Tel. Engineers, Jan. 28, 1886. On the Resistance and Self-induction of Straight Conductors," Phil. Mag., May, 1886, p. 382; Scientific Papers, vol. ii, p. 486. Wied. Ann., xli, p. 400, 1890. Electric Waves, p. 160. less than that amount. Chemically deposited silver failed when the film was so thin as not to be opaque to light. The thickness of this film he places at less than 1/1,000mm. It was probably not 1/10 of that thickness and, moreover, hardly continuous metal. He remarks that the action of the waves scarcely penetrates farther into the wire than does the light which is reflected from its surface. Similar experiments on the screening effect of extremely thin metal leaf are given by Lodge and others. A calculation of the superficial shell effective in the reflection of Hertzian oscillations is given by Poincaré,* who finds the thickness at which the effect is of its amount at the sur face, 1 -- e for frequency, n = 50 × 10°, thickness = 0.002cm; 66 This estimate is probably too high. = J. J. Thomson has treated the "skin"-effect with considerable fulness. In a note appended by him to Art. 690 of the third edition of Maxwellt he obtains as the resistance per unit length of the conductor where the symbols have the meanings given above. This, again, is the same as the values obtained by Rayleigh and Stefan. We can obtain the relative current densities at different depths below the surface for any given frequency in the following way. Stefan has shown that if w be the component of the current in the direction of the axis of 2, and if it does not vary with 2, the equation When the depth to which the action penetrates is small the effect of curvature of the surface may usually be neglected, in which case (4) may be replaced by #Oscillations Électriques, p. 246 and fol. + See also Recent Researches, p. 246 and fol. Sitzungsberichte der Wiener Akad. der Wiss., xcv (II), p. 917, 1887. (5) This is Fourier's well known equation of diffusion, which Lord Kelvin has shown to be applicable to the motion of a viscous fluid, as of closed electric currents within a homogeneous conductor, of heat, of substances in solution, of electric potential in the conductor of a submarine cable; and, indeed, to every case of diffusion in which the substance concerned is in the same condition at all points of any one plane parallel to a given plane. Suppose, now, the periodic current at the surface to be represented by We have to solve (5) subject to the conditions, wherein σ/4πμ. As t increases the condition of affairs approaches a "permanent" state, and then (9) reduces tot e COS dB-cos pt -B sin x dB] (9) 2K Vt At the surface, i. e. when x=0, the maximum value of the 1 current is I. It becomes of this value at a depth e This depth J. J. Thomsont and Poincarés take as the thick- 1 (radian) 57.3° For high frequencies this thickness becomes exceedingly See Byerly's Fourier's Series and Spherical Harmonics, Art. 51. § Oscillations Électriques, p. 252. it was thinner than this "skin"; and if so, what was the critical thickness in any particular case. In the experiments to be described oscillators were used with frequencies approximately 375, 825, 2000, 3200 millions per second, respectively. Substituting these values for n in the above value (11) for x; and taking μ=1, σ=1600, approximately, for copper or gold, 13,500 for platinum and 4,770,000 for electric-light carbon, we obtain the following table: We can obtain an approximate value for the thickness of the "skin" in another way. Suppose the conductor to be a circular cylinder. From Stefan's formula (2) and assuming that, as with steady currents, the resistance is inversely proportional to the area of the section used by the current, the oscillatory current must occupy the R/R'th part of the section. Since this portion is a thin layer next the surface we have which turns out to be precisely the value we obtained before (11). Hertz stated that, as far as he could observe, the nature of the metal out of which his resonator was formed had no influence upon the phenomena, but experiments by Bjerknest did not confirm this conclusion. He found the efficiency of the metals copper, brass, silver, platinum, nickel, iron, to be in the order in which they are here named. By depositing elec *Electric Waves, p. 45. Wied. Ann., xlviii, p. 592, 1893. |