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THESES submitted To the Faculty of Sciences of the University of Paris for the purpose of obtaining the degree of ENGINEER-DOCTOR by Mr. FELDENKRAIS Graduate Engineer, E.E.P. 1st THESIS — On the measurement of very high voltages of electrostatic generators of the Van de Graaff type. 2nd THESIS — Propositions given by the Faculty. Mr. M. PAUTHENIER — President Mrs. I. JOLIOT-CURIE } Examiners Mr. R. LUCAS — — — FOREWORD I venture to ask the Jury’s indulgence on two points: (1) This work was done in Paisley, a small town to the west of Scotland; it was typed on an English machine by an Englishwoman. Thus a certain number of mistakes were added to those for which I myself am responsible. (2) This work has therefore lost some of its topicality. The preparation of this thesis was, in fact, complete toward the end of 1939. The present version is a reconstruction made recently in part from the publications in the Comptes Rendus and in the Journal de Physique et du Radium that I produced in collaboration with our teachers as the work progressed, and in part from scattered documents that I was able to recover. A fallible memory did the rest. Thus the present text has neither the scope nor the careful documentation of the original work. Part of the documentation that I accumulated over several years of work was abandoned along with my belongings—of my own accord—in the following circumstances. I left Bordeaux on 22 June 1940 under orders of mission from the Ministry of National Education, delegated by Mr. Moriolez, Dean of the Faculty of Sciences of Bordeaux. I was ordered to embark at Bayonne for England and to carry three suitcases of secret “Centro” documents which he believed essential to remove from inspection by the enemy, who was at the gates of the city. With the enemy advancing rapidly I was only able to embark at St-Jean-de-Luz on the last boat. The captain’s order was to leave all baggage and embark persons only. I managed, under dramatic circumstances, thanks to the commander of the squadron escorting the ship, to obtain an exception for the three suitcases. Of my effects abandoned on the coast I kept only a briefcase containing documents which, by happy chance, were among the few pieces of baggage brought at the last moment onto the ship after the people had embarked. Professor J. D. Bernal, a devoted friend of my teacher Mr. Paul Langevin (I was a collaborator in Group 4 at Centre carte No. 924) and Lord Suffolk of the British Scientific Mission to the Government of the Republic did everything possible to allow me to carry out my mission despite complications and great material difficulties. Attached: a reproduction of the mission order in question. Paris, 28 October 1945. UNIVERSITY OF BORDEAUX FACULTY OF SCIENCES DEAN’S OFFICE National Center for Scientific Research — — — Bordeaux, 21 June 1940 Mission Order Mr. FELDENKRAIS, Moshe, British subject, collaborator of the National Center for Scientific Research, and his family, will travel to Bayonne on 21 June 1940 in order to embark for England. He will carry documents intended for the Director of Scientific Research and will deposit them at the Embassy of France in London. In the event that he cannot embark, he will join the laboratories group in Toulouse. The Minister of National Education By delegation: The Director, — — — TABLE OF CONTENTS INTRODUCTION ............................................. 1 CHAPTER I Preliminaries .............................. 2 CHAPTER II Review of methods for measuring high voltages .................................... 5 CHAPTER III Spherical-pendulum voltmeter ................ 13 CHAPTER IV Rotating voltmeter .......................... 25 CHAPTER V Electrostatic valve .......................... 30 CHAPTER VI “Method of the work” ........................ 32 CHAPTER VII The use of CO₂ in high-voltage technique .... 37 CHAPTER VIII Conclusions ................................. 40 — — — CHAPTER I PRELIMINARIES The voltages of large power networks are still, for the most part, below 500 kV. One rarely measures the transmission voltages themselves. In insulation tests, voltages on the order of the MV are used. The measurement of voltages on the order of 10 MV has become commonplace in nuclear-physics laboratories. For the physicist, measuring such voltages presents no great difficulty. Indeed, measurement of the energies of accelerated ions, X‑ray spectra, and methods borrowed from electron optics make it possible to obtain fairly precise values of the voltages employed. These methods have the drawback of not being usable for continuous measurements. They therefore most often serve to calibrate a more or less industrial instrument. After the discovery of artificial radioactivity, many physicists called for installations to produce radioactive isotopes. The high voltages required also make possible the production of very penetrating X‑rays. The use of carbon tetrachloride, freon, and other vapors under pressure makes it possible to reduce the dimensions of high‑voltage generating apparatus. The construction of large buildings that were formerly necessary to house such equipment—where this gas is no longer indispensable—still leaves, today, a method employing portable apparatus for producing high voltage. Thus the measurement of very high voltages has become a more or less industrial necessity. Laboratory methods prove impractical: 1. because highly qualified personnel are indispensable, 2. results are known only long after the measurement, 3. measurements are not continuous but made at separate instants. Modern high‑voltage generators are of the Van de Graaff or Pauthenier type. In these devices the voltage is limited by leakage, which increases rapidly with voltage. It is therefore essential that the voltage‑measuring instrument not draw power from the source. If this condition is not met, the instrument does not provide the service required. Most common high‑voltage measurement procedures are impracticable when dealing with a generator with a sensitive output, like the Van de Graaff. Besides the condition that the measuring device should absorb essentially no power, the device must also not increase leakage by its presence. Indeed, nearby conductors increase leakage via corona (aigrettes) or invisible discharge as soon as their surface field exceeds a certain value. If the grounded conductor has a point of radius of curvature \(r\), it should be placed at a distance greater than \(d\) computed as follows\(^2\). The field at the surface of a sphere of radius \(R\) is \(Q/R^2\); that at the tip of the conductor is \(Q/(r d)\). For the latter to be less than that of the sphere, one must have \(r d > R^2\) or \(d > \frac{R^2}{r}\). One sees that \(d\) is very large; if this condition were absolutely rigorous, one would build gigantic buildings needed to house a high‑voltage apparatus. In practice, much smaller distances are tolerated because conditions are more favorable when conductors are not too far apart. Thus measuring instruments must be placed far from the charged electrode and also sufficiently far from the walls. Rejecting methods that satisfy these conditions, only a small number remain to consider: the sphere electrometer used as a pendulum, Pauthenier’s electrostatic valve, Kilpatrick’s rotating voltmeter, and the “method of the work” proposed by ourselves. The construction of the apparatus and a comparative study of these four methods is the subject of the present work. During the development of the Van de Graaff built (1938) … and his collaborators, Mr. Joliot at the laboratories of the École Spéciale des Travaux Publics at Cachan, the generator operating … made possible the discovery of the effect of carbon tetrachloride on the generator voltage in air and oils. This discovery allows a notable increase in voltage. It was quickly followed, as we had foreseen, by the discovery of a whole series of vapors having the same effect, in particular freon. This discovery was made in the course of our work for the present thesis, and that is why we allow ourselves to discuss it in some detail in the last pages. CHAPTER II Review of Methods for Measuring High Voltages The growing popularization of the different methods of high‑voltage measurement is not easy. A. Wolf and K. Drossbach presented an account of the current state of the question. A. Pinkus provided a fairly complete table according to the possibilities of use in continuous service or discontinuous service. It seems preferable and more useful to divide the set of measurement methods, from the physical point of view, into four parts: I. Electrostatic methods. II. Electromagnetic methods. III. Electronic methods. IV. The “method of the work”. I. Electrostatic methods can be subdivided into three series depending on whether the voltage is deduced (a) from the force of attraction, (b) from induced charge, (c) from electrostatic pressure. (a) Force of attraction Abraham voltmeter (Abraham, Villard) “parrot‑tongue” type (Thomson) compressed‑gas type (Fahr Frank) Bénard–Schœner sphere type (Sorensen, Luxor, Feldenkrais) Tallant electrometer “Volumenzuwachs” ellipsoid type (Bjerknes, Thornton) (b) Induced charge rotating voltmeter (Smith, Kilpatrick) (c) Electrostatic pressure work of Pauthenier II. Electromagnetic methods can also be subdivided into three series: (a) continuous current flow, (b) intermittent flow, (c) voltage divider. — — — - 3 - Modern high-voltage generators are often of the Van de Graaff or Pauthenier type. The voltage in these devices is limited by leakages which increase rapidly with it. It is essential, then, that the voltage-measuring device draw no power from the source. If this condition is not satisfied, the instrument does not render the services demanded of it. Most common high-voltage measuring procedures are impracticable when dealing with a generator of sensitive output such as the Van de Graaff. For, apart from the condition that the power absorbed by the measuring device must here be rigorously small, it is also necessary that the measuring device not increase the leakages by its mere presence. Indeed, the proximity of conductors increases leakages by “aigrettes” (or invisible discharge) as soon as their surface field exceeds a certain value. If the grounded conductor has a point with radius of curvature r, it would have to be placed at a distance greater than d, calculated as follows(2). The field at the surface of a sphere of radius R is \(\frac{Q}{R^2}\); that at the point of the conductor is \(\frac{Q}{r d}\). For the latter to be less than that of the sphere, one must have \[ r d > R^2 \quad \text{or} \quad d > \frac{R^2}{r}. \] One then sees that d is very large, and if this condition were absolutely rigorous, gigantic buildings would be necessary to house a high-voltage apparatus. In practice, notably smaller distances are tolerated because the conditions are more favorable when conductors are close to walls. Thus measuring instruments must be placed far from the charged electrode and also sufficiently far from the walls. Rejecting the methods that do not satisfy the conditions listed above, there remain only a few methods to consider. Among these are the sphere electrometer, which we used as a pendulum, Pauthenier’s electrostatic valve, Kilpatrick’s rotating voltmeter, and the work method proposed by ourselves. — — — - 4 - The construction of the instruments and the comparative study of these four methods are the object of the present work. During the development of the Van de Graaff built (1938) by M. Joliot and his collaborators, in the laboratories of the École Spéciale des Travaux Publics at Cachan, the operation of the apparatus led to the discovery of the effect of carbon tetrachloride on the air breakdown voltage and on leakages(1). This discovery allows a notable increase of the voltage. It was rapidly followed, as we explain further on, by the discovery of an entire series of vapors having the same effect, and in particular Freon. This discovery was made in the course of our work for the present thesis; that is why we allow ourselves to speak of it with a few details in the final pages. — — — - 5 - CHAPTER II Review of Methods for Measuring High Voltages A systematic classification of the different high-voltage measurement methods is not easy. A. Jolley(3) and K. Derenwowski(4) have made a report on the current state of the question. A. Pink drew up a fairly complete table based on the possibilities of use in routine service or demonstration. It seems preferable and more useful to divide the whole set of measurement methods, from the physical point of view, into four fairly distinct parts: I. Electrostatic methods. II. Electrokinetic methods. III. Electronic methods. IV. Work method. I. Electrostatic methods can be subdivided into three series, according to whether the voltage is deduced (a) from the attraction force, (b) from induced charge, (c) from electrostatic pressure. (a) Attraction force Abraham voltmeter(5) (Abraham, Villard) ” with “langue de perroquet”(6) (Thomson) ” with compressed gas(7) (Palm, Frank) ” Starke–Schröder(8) ” with spheres(9) (Sorenson, Mestor(10), Feldenkrais) Pellat electrometer ” Tehomyrsohoff(13) [name uncertain] ” ellipsoid type(11) (Bjorknes, Thornton(12)) (b) Induced charge Rotating voltmeter(14) (Mathias, Kilpatrick(15)) (c) Electrostatic pressure Pauthenier valve(16). II. Electrokinetic methods can also be subdivided into three series: (a) continuous flow of electricity, (b) intermittent flow, (c) voltage dividers. — — — - 6 - (a) Continuous flow Ionic-blast voltmeter(17) (Thornton) ” corona-effect voltmeter(18) (Whitehead) ” effluve-tube voltmeter(19) (Palm, Nyiri(20)) ” limited-current voltmeter(21) (Toeppler) (b) Intermittent flow Klydonograph(22) Point gaps ” sphere gaps ” needle gaps ” disk gaps. (c) Capacitive voltage dividers ” with transformers ” with resistances. III. Electronic methods. Braun tube Kerr cell Electron diffraction X-ray spectrum Ion acceleration. IV. Work method. — — — - 7 - Electrostatic Methods (a) Attraction force The force of attraction or repulsion that appears between two surfaces subjected to a potential difference was used from the outset to measure the voltage between these surfaces. The electrical energy stored in the capacitor formed by the two surfaces is given by \[ \mathcal{E} = \tfrac{1}{2} C U^{2}. \] C being the capacitance of the system, U the potential difference assumed constant between the two electrodes. It is known that if \(dx\) denotes an infinitely small translation of one of the plates along a direction \(x\), the projection on \(Ox\) of the electric force \(F\) acting on the plate is \[ F = \frac{\partial \mathcal{E}}{\partial x} = \tfrac{1}{2} U^{2}\,\frac{\partial C}{\partial x}. \] One sees that the attraction force is proportional to the square of the voltage between the surfaces. It suffices to have geometrically simple surfaces in order to deduce the voltage in absolute units. This is the principle on which the various electrostatic electrometers or voltmeters giving the voltage in practical units are based. Abraham–Villard voltmeter One of the oldest of these instruments is that of Abraham–Villard(5). It consists of two circular electrodes with rounded edges placed concentrically face to face on insulating frames—porcelain up to 50 kV, bakelite rods for higher voltages. These devices generally carry a scale on the frame allowing the electrodes to be separated or brought closer so as to adapt their spacing to the voltage to be measured. At the center of one of the electrodes there is a small disk with turned-down edges that forms the moving assembly. It is this disk which, attracted by the opposite electrode, advances while stretching a spring that brings it back to zero when the effect of the voltage is suppressed. The displacements of this small disk, — — — - 8 - which, for a given electrode spacing, depend only on the terminal voltage, are transmitted by a system of levers to a pointer indicating the voltage in kV. The advantage of this type of instrument is to give a direct and continuous reading. It measures DC voltages as well as AC voltages, for which it indicates the RMS value. But it is sensitive to stray fields, which limits its use to relatively low voltages (300 kV). Prof. Pauthenier showed that the use of the Abraham–Villard voltmeter meets difficulties above 600 kV, and he considers that the indications of the voltmeter no longer correspond to the above formula from the moment when the air around the apparatus can be considered ionized. Starke–Schroeder voltmeter An appreciable improvement is the Starke–Schroeder instrument(8), in which the moving part is shielded from external fields and performs a rotation read by means of a sighting tube. It is built for 400 kV. — — — - 9 - Deyriale(29) proposed adding a spherical capacitor, and Watanabe(30) a compressed-air capacitor, to reduce the influence of stray fields. Sphere voltmeter — see Chapter III Electrometers In principle, electrometers give the value of the voltage in absolute units by making use of the attraction force that appears between the two electrodes. They therefore include a balance beam. Apart from many precision devices designed for the research laboratory—such as the Pellat electrometer and others—one must mention the compressed-gas plate electrometer reported by Palm(31). The apparatus is contained in an iron tube filled with compressed gas, and the attraction force exerted on the moving assembly is balanced by a weight, which makes it possible to determine it in absolute units. The instrument is more precise than the electrostatic voltmeter but takes longer to operate. It is built for 300 kV. The voltage is deduced as follows: the balancing weight provides a force \(mg\) equal to the attraction force acting on the surface \(S\) on which there is an electric surface density \(\sigma\), thus \[ mg = 2\pi\,\sigma^{2}\,S, \] and the field between the two attracting surfaces separated by \(d\) is \[ E = \frac{V}{d} = 4\pi\sigma, \] hence \[ V = d\,\sqrt{\frac{8\pi\,mg}{S}}. \] Tehomyrsohoff electrometer Tehomyrsohoff proposed(13) balancing the attraction force on the moving assembly by a coil traversed by a current. Knowing the characteristics of the coil, it suffices to measure the current that passes through it in order to know the attraction force in electrical units. In reality the system is doubled: two pairs of electrodes and two pairs of coils are placed on the same balance beam. It is shown(7) that this system has the advantage of adjusting the equilibrium of the system by the sole adjustment of the current in the coils. — — — - 10 - The measurement is more rigorous than in the preceding instrument, where a manual intervention is necessary to obtain it. The apparatus is enclosed in a compressed-gas box at a pressure of about 75 atmospheres. The manipulations are simpler than in the preceding case and the precision better, but its cost is very high. The cited author indicates a precision of 0.05% for 300 kV. Ellipsoidal electrometer The measurement method used by Thornton(12)(32) is related to Bjorknes(11). A light metal ellipsoid, about the size of a match, is suspended from a very long wire. The ellipsoid is placed between two plane electrodes, circular with rounded edges, subjected to the voltage to be measured. The longitudinal axis of the ellipsoid is oriented in the direction of the uniform electric field. Under the action of this field the ellipsoid is subjected to a torque that sets it in motion. It is shown(13) that the number of oscillations of the ellipsoid in one second \(f\) is proportional to the field, and the driving torque to its square. The voltage is deduced from the following formula: \[ E = \frac{U}{d} = K\,\sqrt{n^{2} - n_{0}^{2}}. \] in which \(E\) is the field intensity, \(U\) the voltage, \(d\) the distance between electrodes, \(n\) the number of oscillations per second of the ellipsoid placed in the field, and \(n_{0}\) the number of oscillations of the ellipsoid outside the electric field. The apparatus gives RMS values of voltages in absolute units with a precision of 1/1000, according to the authors who used this method to measure the ratio between electrostatic and electromagnetic units. This apparatus is generally used as a voltmeter, the constant of the instrument being determined once and for all with precision. It is known, moreover, under the name “ellipsoid voltmeter.” It can be used for very short-duration voltages and currents. Under the action of an instantaneous voltage, the ellipsoid will have to — — — - 11 - of an angle that is a function of the product of the driving torque and the time during which it is applied. By determining the duration of the impulse, one can deduce the voltage and the current, knowing the characteristics of the circuit. (b) Induced charge, see Chapter IV. (c) Electrostatic pressure, see Chapter V. — — — - 12 - II Electrokinetic Methods (a) Continuous Flow Ionic-blast voltmeter(17) A heated electrode forms one branch of a Wheatstone bridge. The presence of a high voltage produces an “ionic wind” that cools the heated wire; its resistance changes and the bridge is no longer balanced. The deflection of the bridge galvanometer is used to indicate the voltage. The system is especially sensitive for indicating the voltage at which the cooling begins to act. This device is also used to measure alternating voltages, but obtaining precise results becomes delicate. Corona-effect voltmeter(18) The appearance of corona discharge on a wire stretched inside a cylinder depends on the voltage and the pressure according to Paschen’s law. To measure a voltage, the pressure is lowered until the discharge appears. It can be detected visually, or more precisely by a galvanometer in series with a battery, this circuit being open inside the cylinder. When the discharge triggers, the air is ionized and the break becomes conductive, and the triggering is observed on the galvanometer. Effluve tube(19,20) A tube filled with a rare gas, neon or another, lights at a very constant voltage. Palm and Nyiri used such tubes lighting at about 200 volts by mounting them in parallel with a variable capacitor which itself served as a voltage divider. The capacitor graduations can be made in kV. Limited-current method(21) Toepler studied transitions from one form of discharge to another in ambient air. He measured the limiting current intensity and the corresponding voltage. This method is used only in the laboratory. (b) Intermittent Flow Klydonograph(22) A metallic point placed on an insulating plate covering a — — — - 13 - a metal plate make it possible to observe partial discharges at the point in the form of “aigrettes.” The shape and dimensions of the aigrettes, often recorded on photographic plates, give an idea of the order of magnitude of the voltage between the point and the metal plate. Sphere gaps(33,34), needle gaps, disk gaps A fairly complete study is published by Thornton(25). The International Electrotechnical Commission (I.E.C.) has chosen sphere gaps as the standard method for measuring industrial high voltages, but there does not yet exist a general and infallible absolute method that could serve for calibrating the other methods. This high-voltage measurement method in fact presents many disadvantages and sources of error. 1. Peek(35) showed that a much higher voltage is necessary to produce a discharge when the voltage is applied for a very short time than when it is applied for a long time. (With point or needle gaps the difference can be from simple to double). 2. The voltage is deduced from a more or less empirical formula (Peek’s formula). Calibrations carried out in different laboratories give differences of about 10%. Thus, when in doubt, one still prefers Peek’s formula. 3. The measurement is discontinuous; each reading requires a special maneuver changing the sphere spacing up to breakdown, which is generally irregular and leads to repeating the maneuver, which is tedious—and more seriously, the phenomenon is disturbed. 4. The inception can be suppressed by polishing the spot where it persistently occurs. For precise measurements one must repeat them until the gap no longer has persistent preferred points. 5. When the distance between the spheres is greater than their diameter, the gaps introduce a new perturbing factor that can no longer be neglected, so that generally one does not exceed a distance of one diameter. — — — - 14 - Noting further the dependence of these measurements on temperature, barometric pressure, humidity and ionization, and the need for large spaces, one is led to reserve this method for insulation tests and other industrial methods, for it is a simple and robust device. The new prescriptions of the International Electrical Congress, 1935 are [crossed-out word] in the publication I.E.C. No. 52. The new rules of use in America are published in E.E. 1936, p. 783. (c) Voltage Dividers Voltage-divider devices are in common use in industrial practice. Voltages on the order of 250 kV do not present great difficulties. These devices obviously require indicators in order to read the voltage. Capacitive dividers Capacitors with very low losses are needed. It is also necessary that the same current, strictly speaking, pass through all the elements in series. The use of a capacitive divider for measurements of a DC voltage is very difficult. The voltage tends to divide according to the resistance of the elements when the voltage reaches the steady (DC) value. Resistive dividers(36) With alternating current and above 150 kV, the capacitance of the winding, which increases with the dimensions and the volume of the insulation, takes values that must be taken into account. Moreover, because the winding must be weakly inductive, it becomes very costly. M. Joliot and M. Savol built a 7-meter-long tube filled with a mixture of alcohol and xylol, with two platinum electrodes separated by 15 cm on the side connected to earth. One twentieth of the voltage was supposed to appear between the two electrodes. It was difficult to maintain the mixture uniform over the whole length of the tube. There were also leakages on the dehydrator, and the device was abandoned without insistence. — — — Formula in which \(B\) denotes Kerr’s constant. Knowing \(B\) and \(\ell\) and measuring \(\varphi\), one can therefore use a setup of this kind to measure the electric field \(E\). \[ \frac{\varphi}{\pi} = \frac{\delta}{\lambda} = B\,\ell\,E^{2}. \] — — — - 15 - III Electronic Methods(37) It is impossible to enumerate all electronic methods, even briefly, without exceeding the modest limits of our work. The deflection of an electron beam by an electric and magnetic field has given rise to a great number of devices allowing the evaluation of the speed of the electrons in the beam, and thereby the voltage that produced this speed or generated the deflection. Let us nevertheless note a few methods used directly for voltage measurement, or rather for calibrating measuring instruments with linear characteristics. Braun tube This is the prototype of the cathode-ray tube. By applying an electric field one obtains a deflection of the beam from which one deduces the speed of the electrons. The voltage is deduced in the following way: \[ \text{[formula illegible in the photo]} \] \(V\) being the voltage, \(e\) the charge of an electron \((1.60226 \pm 0.00015)\cdot 10^{-20}\) e.m.u. C.G.S., \(m_0\) its rest mass \((9.116 \pm 0.002)\cdot 10^{-28}\,g\), \(C\) the speed of light, and \(v\) the electron speed. Kerr cell Given an insulating liquid between parallel electrodes, the application of an electric field generally makes it birefringent; its neutral lines are respectively parallel and perpendicular to the electric field \(E\). If one makes a light beam traverse the field of length \(\ell\), parallel to the plates and linearly polarized at 45° to the electric field, the emerging beam is generally elliptically polarized, the axes of the ellipse being at 45° to the electric field. It is easy to measure, with a quarter-wave plate, the ratio \(\frac{b}{a} = \tan\varphi\) of the axes of the ellipse. Now if one calls \(\delta\) the optical path difference between the \(\varepsilon\) compo- [← see] Electron diffraction An electron beam undergoes diffraction when passing through a thin metal sheet. By recording the diffraction rings on a film(28), one determines the wavelength and deduces from it the voltage through which the beam has passed. — — — - 16 - X-ray Spectrum X-rays are emitted by any target struck by sufficiently fast electrons. The intensity of the radiation increases with the number of atoms in the target. The hardness of the radiation increases with the speed of the electrons striking the target. Thus the hardness of the radiation—or more precisely the minimum wavelength found in the radiation—corresponds to the greatest speed acquired by the electrons; and this speed is a function of the potential difference between the electron source and the target. When fast electrons strike a target, two radiations are emitted. One has a continuous variation of wavelength whose extent and intensity are functions of the potential difference acting on the electrons along their path toward the target. The lower limit of the wavelengths depends on the potential drop along the path of the electrons. The other radiation has well-defined wavelengths characteristic of the target. This radiation appears only above a certain value of voltage and is otherwise independent of it. It is the smallest wavelength of the continuous spectrum that is used for voltage measurement. According to the Duane and Hunt law(38), \[ V e = h\,\nu_{\max} = \frac{h c}{\lambda_{\min}}. \] \(V\) being the potential difference applied to the tube, and \(\nu_{\max}\) and \(\lambda_{\min}\) the frequency and wavelength of the smallest wavelength observed. The wavelength is determined in absolute value in the most precise way by the diffraction spectrum of a tangent grating of geometric period \(d\). Reinforcements are obtained in directions \(\varphi\) according to the law \[ n\,\lambda = d\,(\cos\theta - \cos\varphi) \] \(\theta\) being the angle of incidence and \(\varphi\) the diffraction angle, \(n = 0, \pm 1, \pm 2, \pm 3\), etc. \(\theta\) and \(\varphi\) are very small. Since cosines near unity vary very slowly, \(\varphi\) is appreciably different for different spectral orders despite the enormous difference between \(d\) and \(\lambda\). Howe’s technique(38a) makes it possible to obtain very fine and well-defined lines. — — — - 17 - Overview Among the methods reviewed, only the following were then actually used for the direct measurement of voltages exceeding one million volts, namely: the sphere voltmeter (Mester), the ellipsoid voltmeter (Thornton), sphere gaps, and the klydonograph. All the other methods require capacitive or resistive voltage dividers, of which only a part—about 250 kV—is measured directly. Two exceptions to this generalization are the Abraham–Villard voltmeter (500 kV) and Starke–Schroeder (500 kV). For the direct measurement of much higher voltages exceeding one million volts—and especially in the case of apparatus such as the Van de Graaff, where gaps would discharge the apparatus at the very moment of interest and the klydonograph has no practical usefulness—one is led to consider only the sphere voltmeter, the rotating voltmeter, and the work method. All these methods can be used while the apparatus is in operation without changing its functioning in any way. The electrostatic valve is a useful instrument for absolute calibration at high voltages. — — — - 18 - CHAPTER III Spherical torsion electrometer Description of the apparatus (Fig. 1) An aluminium sphere (B) into which a metal rod (D) is screwed, carrying a small weight (P) whose displacement makes it possible to balance the assembly on the suspension fiber (f), constitutes the essential part of the apparatus. The free end of the rod is extended by a silk thread placed on the rim of a bicycle wheel (C); a box (T) is suspended at the end of this thread. The quantity of water necessary to balance the apparatus with the suspension fiber (f) vertical, together with the weight of the box and its attachments, represents the attractive force exerted between spheres A and B.nd its attachments, represents the attractive force exerted between spheres A and B. To facilitate the damping of oscillations about the equilibrium position, the box (T) is placed inside another one of slightly larger diameter (H). To make the damping more effective, the box (T) is extended by a sheet-metal cylinder soldered to its base so as to increase the friction surfaces during the motion of the boxes. This is the classical device used by P. Curie in his balance. The bicycle wheel is mounted on its axle and carried by its fork, itself mounted on a telescopic stand allowing the rod carrying the sphere and the weight-supporting thread to be adjusted to the horizontal. An index fixed to the screw used to tighten the small adjusting weight P gives the displacement from the equilibrium position. In practice one tries to bring the index to the zero mark in order to keep the spacing between spheres A and B at the distance for which the calculations were made. To connect sphere B to ground, a very fine and flexible wire is clamped to the rod with the clamping screw of weight P; it is then unwound over a lateral guiding arm and runs down along the telescopic stand to a good earth. Experience has shown that an aluminium sphere 0,2 mm thick was too light and too sensitive. The amplitudes were too large and too abrupt. The attractive force was in fact about 750 grams and we — — — - 19 - decided to increase the weight of the whole to about 5 kilograms in order to obtain oscillations sufficiently slow and of reasonable amplitude, allowing easy and reliable readings. A shortening of the pendulum length had the same [continuation illegible in the photo]. Study The attractive force that appears between a charged sphere and another metallic sphere connected to ground is proportional to the square of the applied voltage. It was Lord Kelvin, i.e. William Thomson(6), who indicated the best procedure for finding the factor of this proportionality, which is a function of the diameters of the spheres and of their mutual distance. Here are the theoretical conditions stemming from electrostatics that make it possible to (formulate and solve) the force of attraction between the spheres. A point charge (q), far from any other body, produces at a point (S) at distance (r) a potential V such that \[ V = \frac{q}{r} \] This potential becomes zero for \(r = \infty\). This is the only condition that the potential equation must satisfy in this case. If the charge (q) is placed near a metallic sphere connected to ground, the potential equation must also satisfy the additional condition of becoming an equipotential surface of zero potential at every point (S) on the sphere. If the charge (q) is placed near an isolated conducting sphere, the conditions to satisfy are that the potential at every point (S) on the sphere is constant and that the total charge induced on the sphere is zero. Let us consider a sphere (B) with center O and a point P outside the sphere, where a charge (q) is placed. On the line OP (see Fig. 2) joining point P to the center O of the sphere, one can find a point P' such that its distance \(d\) from the center is \[ \frac{a}{d} = \frac{f}{a} \quad \text{that is} \quad d = \frac{a^{2}}{f} \] Thus one has two triangles PSO and P'S'O in which — — — - 20 - \[ r = \sqrt{a^{2} + f^{2} - 2 a f \cos \theta} \] and \[ r' = \sqrt{a^{2} + d^{2} - 2 a d \cos \theta} \] replacing \(d\) by its value \(\frac{a^{2}}{f}\) \[ r' = \sqrt{a^{2} + \frac{a^{4}}{f^{2}} - 2 \frac{a^{3}}{f} \cos \theta} = \frac{a}{f}\sqrt{a^{2} + f^{2} - 2 a f \cos \theta} \] hence \(r' = \frac{a}{f} r\), or again \(\frac{1}{r} = \frac{a}{f}\,\frac{1}{r'}\) the ratio \(\frac{a}{f}\) being constant, this remains true for any point S on the sphere. The sphere being supposed connected to ground, its potential is zero, but one finds an induced charge on the sphere due to the presence of the charge (q) at P. The induced charge on the sphere distributes itself in such a way that the potential at every point S on the sphere is equal to zero. It is impossible to find this distribution, but one can find the expression of a charge (q') placed at a point inside the sphere that would give a final result equivalent to this distribution. Indeed, let us suppose a charge \[ q' = - \frac{a}{f}\, q \] placed at point P' inside the sphere, defined as above, that is to say \[ \frac{1}{r} = \frac{a}{f}\,\frac{1}{r'} \] the expression of the potential at a point S on the sphere will be \[ V = \sum \frac{q}{r} = \frac{q}{r} + \frac{q'}{r'} = q\left[\frac{1}{r} - \frac{a}{f}\,\frac{1}{r'}\right] = q\left[\frac{a}{f}\,\frac{1}{r'} - \frac{a}{f}\,\frac{1}{r'}\right] = 0 \] A charge \(q' = - \frac{a}{f} q\) placed at P' at \(d = \frac{a^{2}}{f}\) from the center of the sphere is therefore indeed equivalent to the true distribution of the charge induced on the sphere by the charge (q) at P. P' is the electric image of P and the expression above completely defines the field outside the grounded sphere. Thus the field outside a grounded sphere, due to it and to the point charge placed in its vicinity, is identical to the field — — — - 21 - produced by the charge itself and its electric image \(q'\) placed at P'. This reasoning can be extended to the case of an isolated sphere A at potential V whose center is at P, instead of a point charge. One knows, in fact, that the field outside an isolated sphere of capacitance C at potential V is equivalent to the field that would be produced by a charge \(q_{1} = C V\) placed at its center. If the radius of sphere A is equal to a, \(q_{1} = a V\). Let us now suppose two identical spheres A and B, A being isolated and at potential V, B being grounded, their radii being a, and C being the distance between the two centers; the field due to \(q_{1} = a V\) is distorted by the presence of B connected to ground, which always keeps zero potential by accumulating a certain charge on its surface. We saw above that this charge can be replaced by a point charge \(q_{2} = - \frac{a}{C} q_{1}\), placed at \(d_{2} = \frac{a^{2}}{C}\) from the center of sphere B; it restores zero voltage at its surface. It is evident that the presence of this charge on the surface of B in turn influences the distribution on sphere A, and this influence is equivalent to the action of a new point charge \(q_{3}\), which is the electric image of \(q_{2}\), and which must be placed at \(d_{3} = \frac{a^{2}}{C - d_{2}}\) from the center of A. By this reasoning one sees that the real distribution of charges on the two spheres can be replaced by a double series of images, one in sphere A with the sign of \(q_{1}\), the other in B with the sign \(-q_{1}\), at distances \(d_{1}, d_{2}\), etc. from the respective centers of the spheres. The general term of these series is: \[ q_{n} = \frac{a}{C - d_{n-1}}\, q_{n-1} \] \[ d_{n} = \frac{a^{2}}{C - d_{n-1}} \] The attraction between the two spheres is equal to the sum of the forces exerted between each of the charges placed in sphere B and all the charges of A. The force exerted between two charges \(q_{n-1}\) and \(q_{n}\) separated by the distance \(f\) is \[ F = \frac{q_{n-1} \times q_{n}}{f^{2}} \] The total force will then be \[ F = \sum_{(n)}^{\infty}\sum_{(n-1)}^{\infty}\frac{q_{n-1} \times q_{n}}{(C - d_{n-1} - d_{n})^{2}} \] — — — - 22 - The distance (C) between the centers of the spheres being necessarily \(\ge 2a\) (a being the radius of the spheres), the n-th term is much smaller than the (n−1)-th and the series obtained is rapidly convergent. The result of the calculations is written in the convenient form \[ F(\text{dynes}) = 0,1375\, S\, V^{2} \] (\(V\) in electrostatic units) S is a factor depending only on the interval between the spheres. In practical units one obtains \[ V(\text{volts}) = 9,405\,\sqrt{\frac{F(\text{gr})}{S(\text{cm})}} \] Here is the spacing factor S for two identical spheres of radius \(a = 50\) cm. interval (cm) S 0 \(\infty\) 5 1,13812 10 0,52852 15 0,32917 20 0,23159 25 0,17252 30 0,13696 35 0,11082 40 0,09274 45 0,07720 interval (cm) S 50 0,06592 55 0,05693 60 0,04965 65 0,04363 70 0,03865 75 0,03441 80 0,03084 85 0,02775 90 0,02509 95 0,02278 100 0,02075 Calculation of the attraction forces between two spheres A and B of radius \(a = 50\) cm, separated by 30 cm. The distance between centers is therefore \(50 + 50 + 30 = 130\) cm. Sphere A at potential V Sphere B at potential 0 \(q_{1} = aV\) \(d_{1} = 0\) cm \(q_{2} = -0,385aV\) \(d_{2} = 19,2\) cm \(q_{3} = 0,174aV\) \(d_{3} = 22,6\) cm \(q_{4} = -0,086aV\) \(d_{4} = 23,2\) cm \(q_{5} = 0,0378aV\) \(d_{5} = 23,4\) cm \(q_{6} = -0,0177aV\) \(d_{6} = 23,45\) cm \(q_{7} = 0,00833aV\) \(d_{7} = 23,5\) cm \(q_{8} = -0,00393aV\) \(d_{8} = 23,55\) cm \[ F = \frac{q_{1}q_{2}}{(130 - d_{2})^{2}} + \frac{q_{3}q_{2}}{(130 - d_{2} - d_{3})^{2}} + \frac{q_{5}q_{2}}{(130 - d_{2} - d_{5})^{2}} + \frac{q_{7}q_{2}}{(130 - d_{2} - d_{7})^{2}} + \cdots \] — — — - 23 - \[ + \frac{q_{1}q_{4}}{(130 - d_{4})^{2}} + \frac{q_{3}q_{4}}{(130 - d_{4} - d_{3})^{2}} + \frac{q_{5}q_{4}}{(130 - d_{4} - d_{5})^{2}} + \frac{q_{7}q_{4}}{(130 - d_{4} - d_{7})^{2}} + \cdots \] In the case where the spheres have different radii, sphere A being of radius a and sphere B of radius b, the preceding series are written: \[ q_{1} = aV \qquad d_{1} = 0 \] \[ q_{3} = \frac{a}{C - d_{2}}\, q_{2} \qquad d_{3} = \frac{a^{2}}{C - d_{2}} \] \[ q_{5} = \frac{a}{C - d_{4}}\, q_{4} \qquad d_{5} = \frac{a^{2}}{C - d_{4}} \] \[ q_{2} = -\frac{b}{C - d_{1}}\, q_{1} \qquad d_{2} = \frac{b^{2}}{C} \] \[ q_{4} = -\frac{b}{C - d_{3}}\, q_{3} \qquad d_{4} = \frac{b^{2}}{C - d_{3}} \] \[ q_{6} = -\frac{b}{C - d_{5}}\, q_{5} \qquad d_{6} = \frac{b^{2}}{C - d_{5}} \] In order that the auxiliary sphere of radius r, connected to ground, does not increase effluves, it must be placed at a certain distance from the charged sphere of radius R. This distance d is related to the distance between the two centers \(f = R + r + d\) (\(f\) being the center-to-center distance), which leads to equal fields on the surface of the spheres. Taking only the first term of the series above, one finds for the field outside r \[ \frac{Q}{r f} \] and for that of sphere R \[ \frac{Q}{R^{2}} \] Thus: \[ r f \ge R^{2} \] \[ r(R + r + d) \ge R^{2} \] hence \[ d \ge \frac{R^{2}}{r} - (R + r) \ge \frac{50^{2}}{25} - (50 + 25) \ge 25 \text{ cm.} \] with \(R = 2r = 50\) cm in our case. We adopted a separation of 30 cm and later 40 cm. — — — - 24 - In the first case \[ V(\text{volts}) = 3,11 \times 10^{4}\,\sqrt{F(\text{grams})} \] and in the second \[ V(\text{volts}) = 3,803 \times 10^{4}\,\sqrt{F(\text{grams})} \] d = 30 cm d = 40 cm F(gr) V(kV) F(gr) V(kV) 1 31,1 1 38,3 4 62,2 4 76,06 9 93,3 9 114,09 16 124,4 16 152,12 25 155,5 25 190,15 100 311,0 81 342,0 225 466,5 100 380,3 400 622,0 169 494,0 625 777,5 225 575,45 900 933,0 300 658 1034 1000,0 400 760,6 500 850 625 950,75 691,4 1000,0