Order No.:
Series

Y Ing. 1945 (2), 40

30 Nov. 1945

THESES

presented

To the Faculty of Sciences of the University of Paris

to obtain

The Degree of DOCTOR-ENGINEER

by

M. FELDENKRAIS

Graduate Engineer, E.S.P.

1st THESIS — On the measurement of very high voltages
from Van de Graaff type
electrostatic generators.

2nd THESIS — Propositions given by the Faculty.

LIBRARY
OF THE UNIVERSITY
OF PARIS
SORBONNE

FOREWORD

I take the liberty of asking the jury's indulgence on two points:

(1) This work was done at Paulliac, a small village in western Scotland; it was typed on an English typewriter by an Englishwoman. Thus a certain number of errors have been added to those for which I am myself responsible.

(2) This work has also lost some of its timeliness. The preparation of this thesis was, in fact, complete by the end of 1939. The present draft is a reconstruction made recently, in part from the publications in Comptes Rendus and the Journal de Physique et du Radium that I made in collaboration with my supervisors as the work progressed; and in part from scattered documents that I was able to recover. A fallible memory has done the rest.

Thus the present draft has neither the scope nor the careful documentation of the original work.

Part of the documentation that I had 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 from the Ministry of National Education, by delegation from M. Merclor, Dean of the Faculty of Sciences of Bordeaux. I had orders to embark at Bayonne for England and to transport there three suitcases of secret documents from the Centre which he deemed essential to keep from inspection by the enemy, who was at the gates of the city.

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 to embark persons only. In dramatic circumstances, thanks to the commander of the squadron escorting the boat, I managed to obtain an exception for the three suitcases.

Of my belongings abandoned on the coast, I kept only a briefcase containing documents that were, by fortunate chance, among the few pieces of baggage brought aboard at the last moment after the passengers had embarked.

I

complications and great subsequent difficulties.

Attached is a reproduction of the mission order in question.

Paris, 26th October, 1945.

UNIVERSITY OF BORDEAUX
FACULTY
OF
SCIENCES
OFFICE OF THE DEAN

National Centre
for
Scientific Research

Bordeaux, 21 June 1940

Mission Order

Mr. FELDENKRAIS Moshé, British subject, collaborator of the National Centre for Scientific Research, and his family, shall proceed to Bayonne on 21 June 1940, to embark for England.
He shall transport documents intended for the Director of Scientific Research, and shall deposit them at the French Embassy in London.
In the event that he cannot embark, he shall join the group of laboratories at Toulouse.

The Minister of National Education
By delegation:
The Director,

71

III

TABLE OF CONTENTS

Page

INTRODUCTION  ...  ...  ...  1

CHAPTER I  Preliminaries  ...  ...  2

CHAPTER II  Review of Methods for Measuring High
Voltages  5

CHAPTER III  Spherical Pendulum Voltmeter  ...  18

CHAPTER IV  Rotary Voltmeter  ...  25

CHAPTER V  Electrostatic Valve  ...  30

CHAPTER VI  Work Method  ...  32

CHAPTER VII  The Use of CO${}_2$ in High
Voltage Technology  37

CHAPTER VIII  Conclusions  ...  ...  40

- 2 -

CHAPTER I

PRELIMINARIES

The voltages of large electrical networks are still mostly below 500 kV. Transmission voltages themselves are rarely measured directly. In insulation tests, voltages on the order of MV are used. The measurement of voltages on the order of 10 MV has become commonplace in nuclear physics laboratories. For the physicist, measuring voltages of this magnitude presents no great difficulties. Indeed, measuring the energies of accelerated ions, the X-ray spectrum, and methods borrowed from electron optics allow fairly precise values of the voltages employed to be obtained.

These methods have the disadvantage of not being practicable for continuous measurements. They are therefore most often used for calibrating more or less industrial apparatus.

After the discovery of artificial radioactivity, many hospitals have set up installations for the production of radioactive isotopes. The high voltages required also allow the production of highly penetrating X-rays.

The use of carbon tetrachloride¹), Freon, and other pressurized vapors allows the dimensions of high-voltage generating equipment to be reduced. The construction of large buildings that were formerly necessary to house such equipment being no longer indispensable, there are today many hospitals possessing portable apparatus for the production of 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 not known until long after the measurement,
3. measurements are not continuous but taken at separate instants.

- 3 -

Modern high-voltage generators are often of the Van de Graaff or Pauthenier type. The voltage in these devices is limited by leakage currents that increase rapidly with voltage. It is therefore essential that the voltage measuring device draw no power from the source. If this condition is not met, the device does not provide the service required of it.

Most common high-voltage measurement procedures are impracticable when dealing with an apparatus with sensitive output like the Van de Graaff. For besides the condition regarding power absorbed by the measuring device, which must here be rigorously observed, the measuring device must also not increase leakage by its presence. Indeed, the proximity of conductors increases leakage through corona discharge 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 need to be placed at a distance greater than d calculated as follows²). The field at the surface of a sphere of radius R is $Q/R^2$, that at the point of the conductor is $Q/rd$. For the latter to be less than that of the sphere, we need:
$$rd>R^2\text{or}d>\frac{R^2}{r}$$

We see then that d is very large, and if this condition were absolutely rigorous, giant buildings would be necessary to house a high-voltage apparatus. In practice, considerably smaller distances are tolerated because conditions are more favorable when conductors are near the walls.

Thus measuring devices must be placed far from the charged electrode and also sufficiently distant from the walls.

By rejecting methods that do not meet the conditions enumerated above, only a few methods remain to be considered. Among these are the sphere electrometer that we have used as a pendulum, Pauthenier's electrostatic valve, Kirkpatrick's rotary voltmeter, and the work method proposed by ourselves.

- 4 -

The construction of the apparatus and the comparative study of these various methods is the subject of the present work.

At the time of commissioning the Van de Graaff built and installed in the laboratory by M. Joliot at the laboratories of the École Spéciale des Travaux Publics at Cachan, the capillary operation of the apparatus led to the discovery of the effect of carbon tetrachloride on the breakdown voltage of air and its consequences¹). This discovery allows a notable increase in voltage. It was quickly followed, as we shall explain later, by the discovery of a whole 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, which is why we permit ourselves to discuss it in some detail in the final pages.

- 5 -

CHAPTER II

Review of Methods for Measuring High Voltages

The methodical classification of the different methods for measuring high voltages is not easy. A. John⁵) and K. Drewinowski⁴) have reported on the current state of the question. A. Pellin has drawn up a still complete table according to the possibilities of use in regular service or demonstration.

It seems preferable and more useful to us to divide the 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 methods.

I. Electrostatic methods can be subdivided into three series depending on whether the voltage is deduced from (a) the force of attraction, (b) the induced charge, (c) the electrostatic pressure.

(a) Force of attraction

Abraham voltmeter⁵) (Abraham, Villard)
"    propagation time⁶) (Thomson)
"    compressed gas⁷) (Pellin Frankel)
"    Starck - Schröder⁸)
"    sphere⁹) (Sorenson, Hnatek¹⁰), Feldenkrais)
Tait electrometer¹¹)
"    Toeppler-Holtzwychaff¹²)
"    ellipsoid¹¹) (Ejkman, Thornton¹²))

(b) Induced charge

Rotary voltmeter¹³) (Mathias, Kirkpatrick¹⁵))

(c) Electrostatic pressure

Boucheron pellet¹⁴)¹⁶).

II. Electrokinetic methods can also be subdivided into three series: (a) continuous flow of electricity, (b) intermittent flow, (c) voltage divider.

- 6 -

(a) Continuous flow

Ionic wind voltmeter${}^{17)}$ (Thornton)
"        corona effect${}^{18)}$ (Whitehead)
"        effluve tube${}^{19)}$ (Palm, Ryall${}^{20)}$)
"        limiting current${}^{21)}$ (Kempster)

(b) Intermittent flow

Klydonograph${}^{22)}$

Point spark gaps
"        sphere gaps
"        needle gaps
"        disc gaps.

(c) Capacitive voltage divider
" " "        transformer type.
" " "        resistance type.

III.  Electronic Methods.

Braun tube

Kerr cell

Electron diffraction

X-ray spectrum

Ion acceleration.

IV.  Work method.

- 7 -

I. Electrostatic Methods

(a) Force of attraction.

The force of attraction or repulsion manifested between two surfaces subjected to a potential difference was used from the beginning for measuring the voltage between these surfaces. The electrical energy stored in the capacitor formed by the two surfaces is given by

$E=\frac{1}{2}C{U}^2$

$C$ being the capacitance of the system, $U$ the potential difference assumed constant between the two electrodes. It is known that if $dx$ designates an infinitely small translation of one of the plates in a direction $ox$, the projection onto $ox$ of the electric force $F$ acting on the plate is

$F=+\frac{dE}{dx}=\frac{1}{2}{U}^2\frac{dC}{dx}$

We see that the force of attraction is proportional to the square of the voltage between the surfaces. It suffices to have simple geometric surfaces to be able to deduce the voltage in absolute units.

This is the principle on which the various electrometers or electrostatic voltmeters giving voltage in practical units are based.

Abraham-Villard Voltmeter

One of the oldest of these devices is that of Abraham-Villard⁵).

It consists of two circular electrodes with rounded edges placed concentrically facing each other on insulating frames, made of porcelain up to 50 kV, of Bakelite rods for higher voltages. These devices generally have a scale on the frame allowing the electrodes to be moved apart or closer together so as to adapt their spacing to the voltage to be measured. At the center of one of the electrodes is a small disc with folded edges that forms the moving element. It is this disc which, attracted by the opposite electrode, advances while tensioning a spring that will return it to zero when the effect of the voltage is removed. The displacements of this small disc,

- 8 -

which for a given electrode spacing are conditioned only by the terminal voltage, are transmitted by a lever system to a needle indicating the voltage in kV. The advantage of this type of device is to give a direct and continuous reading. They measure DC voltages as well as AC voltages, for which they indicate the RMS value. But they are sensitive to stray fields, which limits their use to relatively low voltages (300 kV). Professor Pauthenier has shown that the use of the Abraham-Villard voltmeter encounters difficulties above 600 kV and considers that the voltmeter readings no longer correspond to the above formula once the air around the device can be considered ionized.

Starke-Schroeder Voltmeter

An appreciable improvement is the Starke-Schroeder device⁸) in which the moving part is shielded from stray fields and performs a rotation read by means of a sighting tube. It is built for 400 kV.

- 9 -

Generally.
Deyriale²⁹) proposed adding an auxiliary capacitor and Watan³⁰) a compressed gas capacitor to eliminate the influence of stray charges.

Sphere voltmeter  -  see Chapter III
Electrometers

Electrometers give in principle the value of the voltage in absolute units by weighing the force of attraction manifested between the two electrodes. They thus include a balance beam. Apart from numerous precision devices designed for the research laboratory such as the Pellat electrometer and others, mention should be made of the compressed gas plate electrometer reported by Pauth³¹). The device is contained in an iron tube with compressed gas and the force of attraction acting on the moving element is balanced by a weight, which allows it to be determined in absolute units. The device 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 acts with force $mg$ equal to the force of attraction acting on surface $S$ on which there is an electric density $\sigma$, thus

$mg=2\pi {\sigma }^{2}S$

now the field between the two attracting surfaces separated by $d$ is

$E={\displaystyle \frac{V}{d}}=4\pi \sigma$

hence

$V=d\sqrt{8\pi mg/S}$

Kohlrausch Electrometer
Kohlrausch proposed³⁵) balancing the force of attraction on the moving element by a coil carrying a current. Knowing the characteristics of the coil, it suffices to measure the current flowing through it to know the force of attraction in electrical units. In reality the system is doubled: two pairs of electrodes and two pairs of coils are placed on the same beam. It is shown⁷) that this system has the advantage of adjusting the balance of the system solely by adjusting the current in the coils.

- 10 -

The measurement is more rigorous than in the previous device where manual intervention is necessary to obtain it. The device is enclosed in a compressed gas box at a pressure of about 15 kilograms. The manipulations are simpler than in the previous case and the precision better, but its price is very high. The cited author indicates a precision of 0.01% for 300 kV.

Ballistic Electrometer

The measurement method used by Thornton¹²)³²) is due to Bjerknes³¹). An ellipsoid of light metal the size of a match is suspended on a very long wire. The ellipsoid is placed between two flat, circular electrodes 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. As a result of the action of this field, the ellipsoid is subjected to a torque that sets it in motion. It is shown¹³) that the number of oscillations of the ellipsoid per 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, $n_0$ the number of oscillations of the ellipsoid outside the electric field.

The device gives RMS values of voltages in absolute units with a precision of 1/1000 according to the authors who used this method for measuring the ratio between electrostatic and electromagnetic units.

This device is generally used as a voltmeter, the device constant being determined once and for all with precision. It is known, moreover, as the ellipsoid voltmeter.

It can be used for voltages and currents of very short duration. Under the impulse of an instantaneous voltage, the ellipsoid must

- 11 -

through an angle that is a function of the product of the driving torque and the time of its application. By determining the shock time, one can deduce the voltage and 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 wind voltmeter¹⁷)

A heated electrode constitutes one of the branches of a Wheatstone bridge. The presence of a high voltage produces an "ionic wind" cooling 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 cooling begins to act. This device is also used to measure alternating voltages, but obtaining precise results becomes delicate.

Corona effect voltmeter¹⁸)

The appearance of corona discharge on a wire stretched in a cylinder depends on voltage and 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 broken 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 tube¹⁹,²⁰)

A tube filled with rare gas, neon or other, lights up at a very constant voltage. Palm and Ryall made use of such tubes lighting up at about 200 volts by mounting them in parallel with a variable capacitor which is itself equipped with a voltage divider. The capacitor graduations can be made in kV.

Limiting current²²)

Toepler studied the transitions from one form of discharge to another in ambient air. He measured the intensity of the limiting current and the corresponding voltage. This method is used only in the laboratory.

(b) Intermittent Flow

Klydonograph²¹)

A metal point placed on an insulating plate covering a

- 13 -

metal plate allows one to observe partial discharges in the form of streamers at the point. The shape and dimensions of the streamers, often recorded on photographic plates, give an idea of the order of magnitude of the voltage between the point and the metal plate.

Sphere spark gaps${}^{33,34)}$, needle gaps, disc gaps

A fairly complete study is published by Thornton${}^{23)}$.

The International Electrotechnical Commission (IEC) has chosen sphere spark gaps as the standard method for measuring industrial high voltages, but there is not yet a general and infallible absolute method suitable for calibrating other methods.

This method of measuring high voltages presents indeed many disadvantages and sources of error.

1.  Peek${}^{35)}$ has shown that a much higher voltage is necessary to cause 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 twofold.)

2.  The voltage is deduced from a more or less empirical formula (Peek's formula${}^{34)}$). Calibrations performed in different laboratories give discrepancies of about 10%. Thus, in doubt, Peek's formula is still preferred.

3.  The measurement is discontinuous; each reading requires a special operation modifying the spacing of the spheres until breakdown, which is generally irregular, and a repetition of the operation, which is tedious and, what is more serious, disturbs the phenomenon.

4.  The triggering point can be eliminated by polishing the place where it occurs persistently. For precise measurements, they must be repeated until breakdown no longer has persistent preferred points.

5.  When the distance between the spheres exceeds their diameter, corona introduces a new disturbance factor that can no longer be neglected, so that generally the distance of one diameter is not exceeded.

- 14 -

Noting further the dependence of these measurements on temperature, barometric pressure, humidity, and ionization, and the necessity of large spaces, one is led to reserve this method for insulation tests and other industrial methods, as it is a simple and robust apparatus. The new prescriptions of the Int. Elec. Congress 1935 are indicated in IEC publication No. 52. The new rules for 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 to read the voltage.

Capacitive dividers

Very low-loss capacitors are required. It is also necessary that strictly the same current flow through all elements in series. The use of a capacitive divider for DC voltage measurements is very difficult. The voltage tends to divide according to the resistance of the elements when the voltage reaches the DC value.

Resistance dividers³⁶)

In alternating current and beyond 150 kV, the capacitance of the coil, which increases with the dimensions and volume of insulation, takes on values that must be taken into account. Moreover, since the winding must be non-inductive, it becomes very expensive.

M. Joffé and M. Savel built a 3-meter long tube of an alcohol and xylol mixture with two platinum electrodes 15 cm apart on the grounded side. One-twentieth of the voltage should appear between the two electrodes. It was difficult to maintain the mixture uniform over the entire length of the tube. There were also leaks on the outside and the device was abandoned without insistence.

components of the incident vibration (wavelength $\lambda$) along the neutral lines, one has at the exit of the field:

$$\frac{\phi }{\pi }=\frac{\delta }{\lambda }=B\ell E^2$$

a formula where $B$ designates the Kerr constant. Knowing $B$ and $\ell$ and measuring $\phi$, one can consequently use an arrangement of this kind to measure the electric field $\overrightarrow{E}$.

- 15 -

III Electronic Methods³⁷)

It is impossible to enumerate all electronic methods without immediately exceeding the modest limits of our work. The deflection of an electron beam by an electric and magnetic field has given rise to a large number of devices allowing the evaluation of the velocity of the beam electrons and thereby the voltage that produced this velocity or caused the deflection.

Let us note, however, a few methods used directly for voltage measurement or rather for calibrating measuring devices 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 the velocity of electrons is deduced. The voltage is deduced as follows:

$V_e=\frac{m_0}{2}\times \frac{v^2}{\sqrt{1-\frac{v^2}{c^2}}}$

V being the voltage, e the charge of an electron (1.60226 ± 0.00015) × ${10}^{-20}$ e.m.u. CGS, $m_0$ its rest mass (9.116 ± 0.002) × ${10}^{-28}$ g, C the speed of light and v the velocity of electrons.

Kerr Cell

Given an insulating liquid between parallel electrodes, the application of the electric field generally makes it birefringent; its neutral lines are respectively parallel and perpendicular to the electric field $\overrightarrow{E}$. If a light beam parallel to the plates and linearly polarized at 45° to the electric field is passed through the field of length $\ell$, 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 $\frac{b}{a}=\mathrm{tg}\psi$ of the ellipse axes. Now if we call $\delta$ the path difference between the 2 compo-

Electron Diffraction

An electron beam undergoes diffraction when passing through a thin metal foil. By recording the diffraction rings on a film²⁸) one determines the wavelength and from this deduces the voltage traversed by the beam.

- 26 -

X-ray Spectra

X-rays are emitted by any target struck by sufficiently fast electrons. The intensity of radiation increases with the number of atoms in the target. The hardness of radiation increases with the velocity of electrons striking the target. Thus the hardness of radiation, or more precisely the minimum wavelength found in the radiation, corresponds to the highest velocity acquired by the electrons, and this velocity is a function of the potential difference between the electron source and the target.

When fast electrons strike a target, two types of radiation are emitted. One has a continuous variation of wavelength whose extent and intensity are a function of the potential difference acting on the electrons in their path to the target, the lower wavelength limit depending only on the potential drop along the electron path. The other radiation has well-defined wavelengths characteristic of the target. This radiation appears only above a certain voltage value 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-Hunt law³⁹)

$$Ve=h{\nu }_{max.}=\frac{hc}{{\lambda }_{min.}}$$

V being the potential difference applied to the tube, ${\nu }_{max.}$ and ${\lambda }_{min.}$ the frequency and wavelength of the smallest observed wavelength.

The wavelength is determined in absolute value most precisely by the diffraction spectrum of a tangent grating with 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 angle of diffraction, n = 0, ±1, ±2, ±3 etc. $\theta$ and $\varphi$ are very small. Since the cosines near unity vary very slowly, $\varphi$ is appreciably different for various orders of the spectrum despite the enormous difference between d and $\lambda$. Howe's technique³⁸ᵃ) allows very fine and well-defined lines to be obtained.

Overall summary

- 17 -

Among recent methods, only the following have actually been used for direct measurement of voltages exceeding one million volts, namely: the sphere voltmeter (Foster), the ellipsoid voltmeter (Thornton), sphere spark gaps, and the Klydonograph. All other methods require capacitive or resistive voltage dividers of which only a portion of about 250 kV is measured directly. There are two exceptions to this generalization: the Abraham-Villard voltmeter (500 kV) and Starke-Schäffner (500 kV).

For direct measurement of much higher voltages exceeding one million volts, and especially in the case of apparatus such as the Van de Graaff where spark gaps would discharge the apparatus just at the interesting moment and the Klydonograph is without practical use, one is led to consider only the sphere voltmeter, the rotary voltmeter, and the work method. All these methods can be used during operation of the apparatus without changing anything in its functioning. The electrostatic valve is a useful instrument for absolute calibration at high voltages.

- 18 -

CHAPTER III

Spherical Balance Electrometer

Description of the apparatus (Fig. 1)

An aluminum sphere (B) into which is screwed a metal bar (D) carrying a small weight (P) whose displacement allows the assembly to be balanced on the suspension wire (f) constituting the essential part of the apparatus. The free end of the bar is extended by a silk thread placed on the rim of a bicycle wheel (C); a tared box (J) is suspended at the end of this thread. The quantity of water necessary to balance the apparatus in the vertical position of the suspension wire (f) of the assembly plus the weight of the box and its attachments represents the force of attraction exerted between spheres A and B.

To stabilize the damping of oscillations around the equilibrium position, box (J) is placed in another of slightly larger diameter (H). To make the damping more effective, box (J) is extended by a sheet metal cylinder welded to its base so as to increase the friction surfaces during displacement of the boxes. This is the classic device used by P. Curie in his balance.

The bicycle wheel is placed on its axle and carried by its fork mounted in a telescopic stand allowing adjustment of the height carrying the sphere and the weight-bearing wire horizontally.

An index fixed to the screw for tightening the small adjustment weight P gives the displacement from the equilibrium position. In practice, one seeks to bring the index to the zero mark in order to maintain the spacing between spheres A and B at the distance for which the calculations were made.

To ground sphere B, a very fine and flexible wire is clamped with the tightening screw of weight P to the bar; it is then unwound over a lateral guide arm and descends along the telescopic stand to a ground terminal.

Experience has shown that an aluminum sphere of 0.2 mm thickness was too light and too sensitive. The amplitudes were too large and too sudden. The force of attraction was indeed about 750 grams if we

- 19 -

... were forced to increase the weight of the assembly to about 5 kilos to obtain oscillations sufficiently slow and of reasonable amplitude allowing easy and consistent readings. A shortening of the pendulum length would act in the same way, but would bring the platform with the suspension hook closer to the charged electrode.

Theory

The force of attraction that manifests between a charged sphere and another identical to it but grounded is proportional to the square of the applied voltage. It is Lord Kelvin, still William Thomson²,⁶) who indicated the manner of conducting the calculation to find the factor of this proportionality which is a function of the diameters of the spheres and the distance between them.

Here are the theoretical conditions arising from electrostatics that allow one to formulate and calculate the force of attraction between the spheres.

A point charge (q${}_1$) distant from any other body produces at a point (S) distant by (r) a potential V such that:

$$V=\frac{q_1}{r}$$

This potential becomes zero for $r=\mathrm{\infty }$. This is the only condition that the potential equation must satisfy in this case.

If one places the charge (q${}_1$) near a metal sphere grounded, the potential equation must also satisfy the additional condition of giving an equipotential surface of zero potential at every point (S) on the sphere.

If one places the charge (q${}_1$) near an isolated conducting sphere, the conditions to satisfy are that the potential at every point (S) on the sphere be constant and that the total induced charge on the sphere be zero.

Suppose a sphere (B) of center O and a point P where a charge (q${}_1$) is placed.

On the line OP (see FIG. 2) joining point P to the center of the circumference O, one can find a point P' such that its distance $d$ from the center is

$$\frac{a}{d}=\frac{a}{f}\text{in other words}d=\frac{a^2}{f}$$

We thus have two triangles PSO and P'S'O in which

- 20 -

$r^{\prime }=\sqrt{a^2+f^2-2af\cos \theta }$

or $r^{\prime }=\sqrt{a^2+d^2-2ad\cos \theta }$

replacing $d$ by its value ${\displaystyle \frac{a^2}{f}}$

$r^{\prime }=\sqrt{a^2+{\displaystyle \frac{a^4}{f^2}}-2a{\displaystyle \frac{a^2}{f}}\cos \theta }$

$={\displaystyle \frac{a}{f}}\sqrt{a^2+f^2-2af\cos \theta }$

hence $r^{\prime }={\displaystyle \frac{a}{f}}r$ or again ${\displaystyle \frac{1}{r^{\prime }}}={\displaystyle \frac{a}{f}}{\displaystyle \frac{1}{r}}$

the ratio ${\displaystyle \frac{a}{f}}$ being constant, this remains true for every point $S$ on the sphere.

The sphere being assumed grounded, its potential is zero, but one will find an induced charge on the sphere due to the presence of charge $(q)$ at $P$. The induced charge on the sphere will distribute 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 $(q^{\prime })$ placed at a point inside the sphere that would give a final result equivalent to this distribution.

Indeed, suppose a charge

$q^{\prime }=-{\displaystyle \frac{a}{f}}q$

placed at point $P^{\prime }$ inside the sphere, defined as above; that is to say that

${\displaystyle \frac{1}{r}}={\displaystyle \frac{a}{f}}{\displaystyle \frac{1}{r^{\prime }}}$

the expression of the potential at a point $S$ on the sphere will be

$V=\mathrm{\Sigma }{\displaystyle \frac{q}{r}}={\displaystyle \frac{q}{r}}+{\displaystyle \frac{q^{\prime }}{r^{\prime }}}$

$=q[{\displaystyle \frac{1}{r}}-{\displaystyle \frac{a}{f}}{\displaystyle \frac{1}{r^{\prime }}}]$

$=q[{\displaystyle \frac{a}{f}}{\displaystyle \frac{1}{r^{\prime }}}-{\displaystyle \frac{a}{f}}{\displaystyle \frac{1}{r^{\prime }}}]=0$

A charge $q^{\prime }=-{\displaystyle \frac{a}{f}}q$ placed at $P^{\prime }$ at $d={\displaystyle \frac{a^2}{f}}$ from the center of the sphere is therefore equivalent to the true distribution of the charge induced on the sphere by charge $(q)$ at $P$.

$P^{\prime }$ is the electrical image of $P$ and the above expression 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 electrical 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.

It is known, indeed, 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 $q_1=CV$ placed at its center. If the radius of sphere A is equal to a, $q_1=aV$.

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 $q_1=aV$ is deformed by the presence of B grounded, which always maintains zero potential by accumulating a certain charge on its surface. We have seen 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, which restores zero potential at its surface. It is evident that the presence of this charge on the surface of B influences in turn the distribution on sphere A, and this influence is equivalent to the action of a new point charge $q_3$ which is the electrical 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 we see that the real distribution of charges on the two spheres can be replaced by a double series of images, one in sphere A of the sign of $q_1$, the other in B of 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 with all the charges of A.

The force exerted between two charges $q_{n-1}$ and $q_n$ separated by distance $f$ is

$$F=\frac{q_{n-1}\times q_n}{f^2}$$

The total force will then be

$$F=\sum _{(n)}^{\mathrm{\infty }}\sum _{(n-1)}^{\mathrm{\infty }}\frac{q_{n-1}\times q_n}{(C-d_{n-1}-d_n{)}^2}$$

- 22 -

The distance $d$ 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 calculation can be put in the convenient form

$F$(dynes) $=0.1375V^2$

($V$ in electrostatic units)

$S$ is a factor depending only on the spacing between the spheres. In practical units one obtains

$V$(volts) $=9.405\sqrt{{\displaystyle \frac{F(gr)}{S(cm)}}}$

Here is the spacing factor $S$ for two identical spheres of radius $a=50$ cm.

Calculation of attraction forces between two spheres $A$ and $B$ of radius $a=50$ cm, separated by 30 cm. The distance between centers is thus $50+50+30=130$ 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

$P={\displaystyle \frac{q_1{q}_2}{(130-d_2{)}^2}}+{\displaystyle \frac{q_3{q}_2}{(130-d_2-d_3{)}^2}}+{\displaystyle \frac{q_5{q}_2}{(130-d_2-d_5{)}^2}}+{\displaystyle \frac{q_7{q}_2}{(130-d_2-d_7{)}^2}}+$

-- 23 --

${\displaystyle \frac{q_1{q}_2}{(130-d_1{)}^2}}+{\displaystyle \frac{q_3{q}_4}{(130-d_2-d_3{)}^2}}+{\displaystyle \frac{q_5{q}_6}{(130-d_4-d_5{)}^2}}+{\displaystyle \frac{q_7{q}_8}{(130-d_6-d_7{)}^2}}+\dots$

In the case where the spheres have different radii, sphere A having radius $a$ and sphere B having radius $b$, the previous series are written:

$q_1=aV{d}_1=0q_2=-{\displaystyle \frac{b}{c-d_1}}{q}_1{d}_2={\displaystyle \frac{b^2}{c}}$

$q_3=-{\displaystyle \frac{a}{c-d_2}}{q}_2{d}_3={\displaystyle \frac{a^2}{c-d_2}}{q}_4=-{\displaystyle \frac{b}{c-d_3}}{q}_3{d}_4={\displaystyle \frac{b^2}{c-d_3}}$

$q_5=-{\displaystyle \frac{a}{c-d_4}}{q}_4{d}_5={\displaystyle \frac{a^2}{c-d_4}}{q}_6=-{\displaystyle \frac{b}{c-d_5}}{q}_5{d}_6={\displaystyle \frac{b^2}{c-d_5}}$

For the auxiliary sphere of radius $\gamma$ grounded not to increase corona, it must be placed at a certain distance from the charged sphere of radius $R$.

This distance $d$ is equal to the distance between the two centers $\rho -R-\gamma$ ($\rho$ being the distance between centers) which leads to equal fields on the surfaces of the spheres.

Taking only the first term of the above series, one finds for the field outside $\gamma$
$${\displaystyle \frac{Q}{\gamma \rho }}$$
and for that of sphere $R$
$${\displaystyle \frac{Q}{R^2}}$$

Thus,
$$\gamma \rho \ge R^2$$
$$\gamma (R+\gamma +d)\ge R^2$$

hence
$$d\ge {\displaystyle \frac{R^2}{\gamma }}-(R+\gamma )\ge {\displaystyle \frac{{50}^2}{25}}-(50+25)\ge 25\text{cm.}$$

with $R=2\gamma =50$ cm in our case.

We adopted a separation of 30 cm and subsequently 40 cm.

- 24 -

In the first case
$V(\text{volts})=3.11\times {10}^4\sqrt{F(\text{grams})}$

and in the second
hence
$V(\text{volts})=3.803\times {10}^4\sqrt{F(\text{grams})}$

- 25 -

CHAPTER IV

Rotary Voltmeter

Description of the Apparatus

The apparatus (Fig. 3) is essentially a generator consisting of a rotor R driven by a motor at constant speed, a collector D (rectifier), and a galvanometer G that gives the current generated by the successive discharges of charges induced on the rotor. The latter consists of a hollow cylinder of red copper 333 mm long, 115 mm in diameter and 3.5 mm thick, at each end of which is soldered a ring on which is screwed a Bakelite disc 15 mm thick, forming the bottom. The metal part is then split along the axis into two half-cylinders, annealed to eliminate internal stresses caused by soldering, and reassembled on the Bakelite discs which are fixed on a steel shaft 20 mm in diameter that passes through them, with the aid of two small rings and screws. The cylinder is electrically isolated from the shaft which is driven by the motor.

The collector is made of two brass segments, each connected to a half-cylinder. An insulating material (Bakelite) was incorporated between the segment joints to avoid vibrations of the braided brass brushes. The brushes are mounted on an insulating brush holder that can be fixed in different positions around the axis; this allows finding the brush position for which the current is maximum. The brush holder disc can also slide along the axis to adjust brush pressure.

Kirkpatrick's apparatus had pole pieces that we have removed here. It is shown below, and experience confirms, that the lack of symmetry of the electric field around the rotating cylinder due to the removal of pole pieces does not affect in any way the essential characteristic of the voltmeter, which is linear calibration allowing easy and precise extrapolation. Only the calibration is no longer absolute in the sense that it is valid only for the position and conditions in which it was performed. It must be redone with each displacement of the apparatus and surrounding objects of sufficient size to affect the electric field.

Consider the initial arrangement of KIRKPATRICK where the rotor rotates between two opposite poles A and B.

- 26 -

The perfection of brush contact was checked by short-circuiting the two half-cylinders of the rotor and passing a current from a battery through one brush to recover it on the other through a sensitive galvanometer. The current was the same whether the rotor was rotating or at rest.

The ball bearings of the cylinder were grounded to prevent stray currents from the motor frame from passing through the galvanometer.

With the charged electrode grounded, a current is observed on the galvanometer. This current, due probably to a thermoelectric effect on the brushes, has little influence on voltage measurements. It corresponds indeed only to a voltage of a fraction of a volt.

Theory

One can say that the rotary voltmeter is an electrostatic generator that, rotating at constant speed, produces a current proportional to the electrostatic field in which it rotates, hence proportional to the potential difference producing the field.   ←── See.

The case of the apparatus and the rotor, being grounded, practically always have the same potential regardless of the voltage applied to the terminals of the apparatus. The only coefficients entering into the establishment of the value of the current produced are $C_A$ and $C_B$ which are the induction coefficients of one half $R$ with respect to $A$ and $B$ under voltage (see Fig. 4).

One can therefore write

$$Q_R=C_A{V}_A+C_B{V}_B$$

$V_A$ and $V_B$ being the potentials of $A$ and $B$ relative to ground. Assuming the most general case where all values of the equation are variable, a change of state is expressed by

$$d{Q}_R=\frac{\partial Q_R}{\partial C_A}d{C}_A+\frac{\partial Q_R}{\partial V_A}d{V}_A+\frac{\partial Q_R}{\partial C_B}d{C}_B+\frac{\partial Q_R}{\partial V_B}d{V}_B$$

$$=V_{A}d{C}_A+C_{A}d{V}_A+V_{B}d{C}_B+C_{B}d{V}_B$$

The total increase in positive charge on half $A$ during a half-turn of the rotor is

- 27 -

$${\int }_{Q_1}^{Q_2}d{Q}_R={\int }_{C_{A1}}^{C_{A2}}{V}_{A}d{C}_A+{\int }_{V_{A1}}^{V_{A2}}{C}_{A}d{V}_A+{\int }_{C_{B1}}^{C_{B2}}{V}_{B}d{C}_B+{\int }_{V_{B1}}^{V_{B2}}{C}_{B}d{V}_B$$

a definite integral in which the lower and upper limits refer to the initial and final conditions of the rotor before and after a half-turn.

The integration formula $\int udv+\int vdu=uv$ allows these integrals to be evaluated without knowing anything about the functional relationship existing between potentials and capacitances. This is important because we see that the capacitor charge circulating between the two rotating half-cylinders is a function of the limiting values of the potential applied at the beginning and end of a half-turn and is independent of the intermediate values that may exist between these two limits. We then have, by integrating

(1)  $Q_2-Q_1=V_{A2}{C}_{A2}-V_{A1}{C}_{A1}+V_{B2}{C}_{B2}-V_{B1}{C}_{B1}$

In the particular case where the apparatus is symmetric by construction, $C_{A1}=C_{B2}$ and $C_{A2}=C_{B1}$

hence

$Q_2-Q_1=\mathrm{\Delta }Q=V_{A2}{C}_{A2}-V_{A1}{C}_{A1}+V_{B2}{C}_{A1}-V_{B1}{C}_{A2}$

The average current flowing in a time $\mathrm{\Delta }t$ of one half-turn is

$$I=\frac{\mathrm{\Delta }Q}{\mathrm{\Delta }t}$$

During a complete turn this current is doubled; if we have $\nu$ turns/sec we have

$$I=2\nu \mathrm{\Delta }Q$$

(A)  $=2\nu (V_{A2}{C}_{A2}-V_{A1}{C}_{A1}+V_{B2}{C}_{A1}-V_{B1}{C}_{A2})$

which is valid for any field provided the rotor is in two symmetric parts.

In the particular case where the field is also symmetric (with respect to the rotor) and constant

$V_{A1}=V_{A2}$ and $V_{B1}=V_{B2}$

and formula (A) can be written

$$I=2\nu [C_{A2}(V_{A2}-V_{B1})-C_{A1}(V_{A1}-V_{B2})]$$
$$=2\nu [C_{A2}(V_{A2}-V_{B1})-C_{A1}(V_{A2}-V_{B1})]$$
$$=2\nu (C_{A2}-C_{A1})(V_{A2}-V_{B1})$$

"30"

or, removing the indices that are no longer useful and denoting
$C=C_{A2}-C_{A1}$, and $V=V_{A2}-V_B$

(B) $I=2\nu CV$

$V$ being the potential difference between A and B and
$C=C_{A2}-C_{A1}=C_{B1}-C_{B2}$ the capacitance of the rotor.

Given that the current shown is independent of variations
of potential that occur during the half-turn considered, one can apply
to the electrodes an alternating voltage of frequency $2\nu$, or a variable
voltage of any form provided its frequency is $2\nu$ or a
multiple of $\nu$, and one would obtain a current that will be a function of
the instantaneous potential difference represented by a certain point of
the curve representing the wave in question. The current will always be $2\nu CV$
as with a constant voltage. To be able to trace the entire curve, it
suffices to be able to change the phase in the rotation of the rotor relative to
that of the applied wave.

This latter condition can be realized in two ways:

(1) By mounting the motor of the apparatus so that one can give it
a variable angular rotation around its axis.

(2) By inserting in the motor circuit a phase-changing transformer
or phase shifter.

By removing sphere B, $C_{B1}=0$ while $C_{A2}$, but $C_{A1}$ is
equal to $C_{B2}$. Making these simplifications, one obtains from (1)

$I=2\nu (C_{A2}-C_{A1})(V_{A2}-V_{A1})$
$=2\nu CV$

the current $I$ being always proportional to the voltage. $C$ and $V$ having
numerical values different from $C$ and $V$ in (B).

Calibration

The calibration of the voltmeter was done by two methods: one with
an Abraham voltmeter, the other with an electrostatic valve.

The first method is based on the following experimental fact. In
an atmosphere containing carbon tetrachloride vapors, the spherical
electrode of the generator charged to about its maximum voltage maintains
its charge for a very long time without appreciable losses. We were able, by

- 29 -

Example: thus the following experiment: The sphere charged to about 100 kV,

take a reading on the voltmeter then approach the charged electrode with a tin
sphere 60 cm in diameter grounded. Starting with a spacing of three
diameters and repeating the measurements by approaching one diameter at each
operation without however producing a discharge. Then the operations were
repeated by removing the tin sphere. The voltmeter readings corresponding
to identical positions of the sphere were the same within reading
error. The second reading being equal to the first, the sphere did
not lose measurable charge during the operations which required a good
quarter of an hour.

The sensitivity of the apparatus is high. Placed at 115 cm from a
charged sphere 30 cm in diameter, it gave 0.56 × 10⁻⁶ Amp per kV, which
allows detection of variations of 500 volts with an ordinary galvanometer.

Fig. 5 gives the results of calibration with an Abraham
voltmeter standardized up to 250 kV.

A series of measurements was made in collaboration with M. Kopola
Vigéron at the laboratory of Professor M. Pauthenier. They allow
confirmation of the observation of J. R. Magnus⁵⁹) that the values given by the
tin sphere are slightly too low at the beginning of the voltage scale,
deviating by 8% at the other end.

With the Van de Graaff of the Cachan laboratories one could raise
the voltage up to 1250 kV; the voltmeter would follow variations of one kV
easily.

The second method is described in the following chapter.

- 30 -

CHAPTER V

Electrostatic Valve

The principle of the electrostatic valve is due to M. Pauthenier²),
a thin pellet resting on the charged sphere lifts when the electrostatic
pressure exceeds its weight.

This method was perfected by M. Léopold Vigneron and in one
detail by ourselves at the laboratory of Professor Pauthenier¹⁶),

Description of the apparatus

At the upper pole of the sphere a hole was drilled with a shoulder (see fig. 6) on which rests a spherical plate 28 mm in diameter, in copper. To the cap is soldered a rod 50 mm long sliding without friction in a quartz capillary tube, supported inside the sphere, to ensure a radial and vertical movement of the cap. The rod soldered to
the cap is threaded to receive on its lower end nuts
of different weights. We were able to make a series of weights ranging from 0.120 gr
to 1.750 gr allowing measurement of voltages from 95 kV.

Small openings are made, at suitable places,
in the wall of the sphere to ensure equal air pressure on both sides
of the cap at the moment of its lifting.

As soon as the product of the electrostatic pressure $2\pi {\sigma }^2$ by the surface area of the cap exceeds
the weight $mg$ of the cap, it lifts and its sharp edge introduces
losses decreasing the sphere voltage. The cap falls back onto its seat to
lift again the next instant. Thus it oscillates and maintains the voltage
between two very close limits.

The voltage is deduced from

$$2\pi {\sigma }^{2}S=mg$$
$$4\pi \sigma =\frac{V}{R}$$
$$\frac{V^{2}S}{8\pi R^2}=mg$$

with the values used for $S$, $m$ and $R$

$$V(\mathrm{kV})=280\sqrt{m(\mathrm{gr})}$$

- 31 -

RESULTS

Tests were made in the open air, far from walls and obviously without a ceiling; thus all harmful influence was eliminated.

The rotary voltmeter was first calibrated by an absolute Abraham and Villard voltmeter of precision around 50 kV. This was done after the linearity of readings on the rotary voltmeter was verified experimentally up to the stable voltage limit of the high-voltage generator.

Then, without changing anything in the arrangement of the apparatus and surroundings, a series of simultaneous measurements was made with the rotary voltmeter and the valve.

In fig. 7—the values given by the valve are plotted as abscissas and those of the voltmeter as ordinates; so that one obtains a line substantially coinciding with the bisector from the origin, hence the degree of agreement of the two methods.

This constitutes an absolute calibration of the rotary voltmeter, the absolute voltage data obtained by the electrostatic valve requiring only the measurement of the weight of the cap assembly and the geometric measurements of its surface and the radius of the charged sphere.

As long as one is not too close to the breakdown voltage at the surface of the sphere, that is to say as long as the surface field is substantially less than 30,000 volt/cm, the Furtwängler pellet carefully made is the simplest of absolute calibration methods. The great advantage of this method lies in the fact that calibration is done at the operating voltage of the apparatus without extrapolation. The precision of calibration is thus improved.

- 32 -

CHAPTER VI.

Van de Graaff Machines.

General Considerations

The principle of modern high-voltage generators is known. Electric charges are brought into a metal collector well insulated from ground. These charges are carried into the sphere by a mechanical carrier = belts of insulating material in the Van de Graaff apparatus, insulating dust in the Hochhauser apparatus. The carrier passes through an ionized space where electric charges are torn from the insulating material that was carrying them into the metal collector which charges.

The supply of charges is compensated by the appearance of charges on the walls due to ionic ionization that originates at the tips through which charges penetrate into the collector. Even in toroidal collectors one eventually reaches a voltage where losses begin to increase more rapidly than they can be replaced. The limiting voltage is thus reached. To maintain this voltage, a continuous supply of charges into the collector is needed to compensate for these losses.

The Van de Graaff used has a spherical collector one meter in diameter mounted on a Bakelite cylinder 50 cm in diameter in which two insulating belts 30 cm wide move at 30 m/sec. (see fig. 8).

The motor power necessary to drive the belts was estimated at mechanical values to which was added the additional power necessary to overcome the repulsion between charges on the belts and the total charge of the sphere. At high voltages this power is not negligible compared to mechanical power as one might think at first.

If charge $Q$ is necessary to charge a sphere of radius $R$ to potential $V$, the belts must bring $Q$ units of charge per second to maintain constant voltage. A horizontal parallel slice on the belts of width $a$ placed at distance $x$ from the

- 33 -

center of the sphere would carry a certain charge $qdx$, where $q$ is the charge carried per unit width of the belts (C/m).

The repulsive force experienced by the belt is

$$F={\int }_R^d\frac{Q_1{Q}_2}{x^2}dx=Q_{1}q{(-\frac{1}{x})}_R^d=Q_{1}q\frac{R-d}{Rd}$$

which is indeed a repulsive force because $d$ is the distance from the lower end of the belt to the center of the sphere and is greater than its radius R.

The mechanical power is the product of this force $F$ by the speed of the belt and is only a small part of the total motor power. The efficiency of this type of apparatus is then low for this reason.

This reasoning useful for the purpose we set ourselves only obscured a fundamental fact that is the basis of voltage measurement by the work method. It was only much later when the apparatus was built and operating that we saw the problem in another light.

Indeed, charges raised by the belts from ground potential to sphere potential absorb mechanical work to raise them from one equipotential surface to another, which is, as is known, independent of the path taken by the charges.

In other words, the additional power $W$ is the work expended to raise a certain charge per second, that is, a current $i$ from potential $V_0$ to potential $V$, now

$$W=i(V-V_0)$$

hence

$$V=\frac{W}{i}+V_0$$

$V_0$ is easily measurable and is moreover negligible except for low sphere voltages.

It is therefore sufficient to measure the current entering the sphere and the work supplied for this purpose by the motor to be able to deduce the voltage 40).

Description of the experimental setup.

In order for the method not to be only an indication of the lower limit of the voltage value, the measured current must

- 34 -

be the one that actually enters the sphere. In larger generators, it is not difficult to read the measuring device inside the sphere itself. In the apparatus we operated, the interior of the sphere was occupied by certain parts intended for ion production and their initial condensation, and all the adjustment and remote operation systems necessary to manipulate them, the sphere being at its maximum potential.

The setup designed to exclude leaks through the insulators of the bearings supporting the belt-carrying drums is shown in Fig. 9.

If the supports were perfect insulators, the two microammeters should, with the generator at rest, indicate the same current. In fact, we obtain the curves in Fig. 10, the difference of which is the leakage current.

The apparatus is then started. As the belts present a neutral surface collecting charges at every moment, the corona discharge is facilitated, the readings of microammeter 1 increase and the current through apparatus 2 decreases because the charges carried away by the belts do not pass through it.

To determine the current brought into the sphere, it is grounded through a current measuring device. This gives the total of the charges entering the sphere when its potential is zero. Fig. 11 gives the ratio of this current to the total flow from the wires as a function of the voltage on the corona-discharging wires.

A more precise value for the current is thus obtained. When the sphere is not grounded and is charging, some charges whose absorption is insufficient are probably prevented from reaching the interior of the sphere by the repulsion effect it exerts on these charges. To minimize these losses, the atmosphere inside the generator is dried by circulating air through a dryer filled with P${}_2$O${}_5$.

To measure the additional power absorbed by the motor for the work done in the electric field, we measure the power absorbed with the sphere grounded. We then obtain the power

now, so to speak, at no-load. Then the sphere is allowed to take its normal potential and the power absorbed by the rotor is determined again. The difference between these two powers is the expenditure in the electric field.

The power absorbed in the electric field being relatively low compared to the total power of the motor, we have recourse to the following procedure to increase the precision of the measurements.

The motor was powered by a source of low internal resistance to give a very constant voltage at the motor terminals. A thermal ammeter was used to measure the current absorbed by the motor. The graduations of such a device expand as they are proportional to the square of the current. We were able to obtain an ammeter where the current reading, the sphere being at ground potential, was in the middle of the scale approximately. The reading of the additional current absorbed when the ground connection is removed was then made at the upper end of the scale where the needle moved about 45° to measure 1 amp.

The voltage at the motor terminals being sensibly constant, one can graduate the ammeter in watts directly.

Assuming that the dielectric of the belts in the dried and heated atmosphere is perfect, one obtains the relative error of the voltage measurement:

$$\frac{\mathrm{\Delta }V}{V}=\frac{\mathrm{\Delta }\left(\frac{W}{i}\right)}{\frac{W}{i}}=\frac{i\mathrm{\Delta }W-W\mathrm{\Delta }i}{iW}$$

In the conditions of our measurements $\frac{\mathrm{\Delta }V}{V}\approx 0,04$. An example of a series of measurements is given in Figures 12, 13 and 14.

Critique of the Method

The current arriving at the collector being measured directly by an apparatus located inside the collector itself, the readings were made by light reflection through the collector support; the method is quite precise and very convenient.

- 36 -

Its essential advantage is that its use introduces strictly no disturbance in the collector's field and takes strictly no power from the generator.

The measurement of power absorbed by the rotor can be done with greater precision than we achieved by passing only the motor current through the ammeter—a direct current of opposite sign, but equal to the current absorbed by the rotor when the collector is grounded. One would then have in this device an auxiliary star impedance to measure the magnetizing or excitation current.

The only limitation of the method is that it cannot be applied to variations in potential that are too rapid. The motor driving the belts has inertia, and small, very fast variations are only reflected on the ammeter after a certain delay.

- 37 -

CHAPTER VII

The Teletransmission of Cochem and the Influence
of Voltage on Electrostatic Generators.

The generator built at Cochem for the synthesis of radioactive elements has a spherical collector one meter in diameter supported by a bakelite cylinder 30 cm in diameter, inside which two 30 cm wide belts carry the charges.

With the apparatus operating in air, we obtain under the best conditions a limiting voltage of approximately 650,000 volts.

However, from time to time, for no apparent reason, the slight noise of corona discharges and brushes disappeared, and sparks up to 250 cm in length, corresponding to a much higher voltage, occurred.

The current brought by the belts was found to be decreased (although we were hardly certain of what is normally measured), but the measurement of the additional power absorbed by the motor for the work in the electric field was almost double. Thus, the length of the sparks corresponded well to an increase in voltage and not to a facilitation of the discharge.

We were convinced that the explanation should be sought in a change in the ambient atmosphere, except for hygrometric variations, temperature, and pressure, the influence of which is

insufficient to account for the observed phenomenon.

We will quickly proceed to examine the influence of the vapors of the solvents used to degrease the solid plates used for the hermetic sealing of the vacuum tube for ion condensation. Carbon tetrachloride was, in fact, the cause.

Here is an example taken from a note we published in the Comptes Rendus(40).

I   Flow of charges carried by the belts in microamperes.

II   Power absorbed by the motor, multiplied by its efficiency, to raise these charges from ground potential to that of the sphere; in watts.

III   Sphere voltage V = W/I in kilovolts.

This increase of nearly 100% was obtained by evaporating 1 liter of CCl${}_4$ in the room where the apparatus was located, which measures 500 m${}^3$.

The interpretation of the effect of carbon tetrachloride is not easy. According to Mr. Jolot "the effect is due on the one hand to the presence of heavy molecules giving ions of low mobility and on the other hand to the presence of chlorine having a high electron affinity" 41).

The study by G.M. Kovalenko undertaken to elucidate this phenomenon(41) would justify a revisit of the question. He found that the increase varies from 1.8 to 2.5 times the voltage obtained in air, and that the potentials obtained in a compound atmosphere are equal to the sum of the potentials that would be obtained in the components at pressures respectively equal to their partial pressures in the mixture.

The effect of tetrachloride is at its maximum in the presence of a point and a sphere. This effect is comparable to the experiments of Natterer (1890) on the disruptive distances between electrodes immersed in different gases or vapors. He specifically shows that the disruptive distance for a given voltage is about 10 times smaller in carbon tetrachloride than in nitrogen.

- 39 -

According to the results of W. Kowalski, the capacity of the collector could be considered as consisting of two parts. One is the capacity in air, the other in the gas in question at its partial pressure.
The two capacities would act in a series circuit, leading to a decrease in total capacity and an increase in voltage with the same charge.

Unfortunately, this is nothing more than a way of looking at it.

Besides the harmful effect of inhaling CCl${}_4$, it should be noted that under the influence of discharges, phosgene (COCl${}_2$) is formed, which is an asphyxiant.

CH${}_2$Cl${}_2$ (freon) produces the same effect on the points as CCl${}_4$, but to a notably more marked degree.

- 40 -

CHAPTER VIII

Conclusions

We have removed the pole pieces from the Kirkpatrick voltmeter and have shown that the linearity of the device is preserved. This simplification makes the device more useful for measuring very high voltages. The only disadvantage that results is the necessity of recalibration each time the device itself or surrounding objects are moved.

The spherical pendulum electrometer requires careful damping, otherwise one must take the average between the limiting amplitudes. It is less stable than the rotary voltmeter.

We have made small openings near the electrostatic valve to equalize the pressure on both sides at the moment of its lifting; this allows more consistent readings to be obtained.

The electrostatic valve is a very convenient means for absolute calibration (spherical collectors only) at high voltages of devices giving continuous readings.

The work method that we have proposed always gives the lower limit of the voltage. It is entirely general and applicable to collectors of any shape.

The action of CCl${}_4$, discovered in the course of our work, could be explained as follows. Because of the electroaffinity of chlorine and the low mobility of the CCl${}_4$ molecule—mobility being inversely proportional to the square root of molecular weight—the sphere rapidly becomes enveloped in a layer in which the proportion of CCl${}_4$ molecules predominates. Losses decrease and the voltage rises.

It is important to verify this hypothesis experimentally, for it would establish that a charged electrode separates certain molecules according to their mobility, producing a higher concentration of heavy molecules.

- 41 -

akin to a mirage. One would thus have another industrial means of separating a non-conducting gaseous or liquid mixture into its constituents according to their molecular weights. Certain isotopes could be separated in this manner.

BIBLIOGRAPHY

1) P. Joliot, M. Feldenkrais et A. Lazard. C.R. 202, 1936, p. 291.
2) M. Pauthenier et Mme. M. Moreau-Hanot. C.R. 202, 1936, p. 929.
3) Conf. Intern. 1937, Paris, A. Palm, Session 24 Juin - 2 Juillet.
4) Conf. Intern. 1937, Paris, K. Drewnowski, Session 24 Juin - 2 Juillet.
5) Abraham et Villard. Journ. de Phys. 5 ème série t. 1 Juillet 1911, p. 525.
6) William Thomson, Papers on Electrost. & Magnetism (McMillan & Co.) 1872.
7) A. Palm. S. Techn. Phys. Vol. X, 1920, p. 137.
   A. Palm. Elektr. Z. Vol. 47, 1926, p. 873.
8) H. Starke et R. Schröder. Arch. Elektr., Vol. 20, 1928, p. 115.
   H. Starke et R. Schröder. Arch. Elektr., Vol. 26, 1932, p. 279, D.R.P. 515227.
9) R.W. Sorensen, J.E. Hobson et S. Ramo. Electr. Eng. Vol. 54, 1935, p. 651.
10) Rueter. Elektr. Z. Vol. 55, 1934, p. 833.
11) Bjerkness. Wied. Ann. Vol. 48, 1892, p. 594.
12) W.M. Thornton. J.I.E.E., Vol. 69, 1931, p. 1273.
    W.M. Thornton. J.I.E.E., Vol. 71, 1932, p. 1.
13) Tschernyscheff. Phys. Zeitschr. 1910.
14) A. Mathias. Elektrizitätswirtschaft, Vol. 25, 1926, p. 297.
15) P. Kirkpatrick et J. Miyake. Rev. Scient. Instr. Vol. 3, 1932, p. 1 & 430.
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17) Thornton, Waters & Thomson, J.I.E.E., Vol. 69, 1931, p. 535.
    Thornton, J.I.E.E., Vol. 69, 1931, p. 1273.
    Thornton, J.I.E.E., Vol. 79, 1936, p. 483.
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    Trans, A.I.E.E., 1920, p. 1056.
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    E.T.Z. 1926, p. 873.
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    J.I.E.E., Vol. 69, 1931, p. 409.
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    E. Wilkinson. Elektr. Z., Vol. 54, 1935, p. 627.
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43

26) Trümper. Arch. Elektr. Vol. 26, 1932, p. 562.
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Fig. 1 — Spherical Balance Electrometer
Legend: A = sphere under voltage; B = mobile sphere; C = bicycle wheel; D = metal bar; E = lateral guide; f = suspension wire; G = clamping ring; H, J = damper (liquid weight); P = balancing weight; scale −2, 1, 0, 1, 2, +; telescopic foot; silk thread; fork.

Fig. 2 — Geometry of Electrical Images
Legend: P = position of charge (q); P′ = position of image (q′); O = center of sphere B; S = point on surface; a = radius; f = distance OP; r, r′ = distances PS and P′S.

Fig. 3 — Rotary Voltmeter
(Photograph of the apparatus)

Fig. 4 — Rotor Positions of the Rotary Voltmeter
Legend: A, B = electrodes (+); R = rotor (divided cylinder); G = galvanometer.
Position A₁ and B₂ after ½ turn; Position B₁ and A₂ after ½ turn.

Fig. 7 — Calibration Curve: Rotary Voltmeter vs. Electrostatic Pressure
Axes: Ordinate = Voltage by rotary voltmeter (0–400 kV); Abscissa = Voltage by electrostatic pressure (0–400 kV).

Fig. 5 — Calibration of the Rotary Voltmeter
Scales: (1 mm = 1 kV) Voltage; (1 mm = 0.5 × 10⁻⁸ A) Current.
K = 0.56 × 10⁻⁸ Amp per Kilovolt.

Fig. 6 — Electrostatic Valve
Legend: spherical cap (valve); quartz capillary tube; weight nut (suppl.).
Note: fig. 7 after fig. 4.

Fig. 8 — Van de Graaff Generator Schematic
Legend: metal sphere; platinum wire (corona charger); insulating column; one of two belts; Motor; 13 kV / 110 V circuit; (+) and (−) = polarities; μA = microammeter; platinum wire.

Fig. 9 — Current Measurement Circuit
Legend: Kenotron (p) = rectifier tube; U′ = voltage; A (1, 2) = ammeters; insulating supports; one of the belts.

Fig. 10 — Charges Carried by Belts and Leakage from Supports
Upper graph: Microamp. readings (1) and (2) as a function of belt speed.
Normal belt speed; U = 11.0 kV; U_max = 11.0 kV; I_max = 10 Amp.
Lower graph: Effect of belt speed; U = 9.0 kV; U_max = 11.0 kV; I_max = 1.0 Amp.

Fig. 11 — Ratio of Charge on Sphere to Current Discharged
Title: Ratio of charge existing on sphere to established current discharged by wires at the same instant.
Axes: Ordinate = 0–50%; Abscissa = Voltage U′ (0–100).
U = 1.18 U′.

Fig. 12 — Wire Discharge by Corona Effect (I)
Axes: Ordinate = Current I (0–1200 μA); Abscissa = Voltage (in volts).
Curves: Under load; In operation; With CCl₄; At rest / idle.
Scales: I: 1 mm = 1×10⁻⁸ Amp; U: 1 mm = 1 volt; τ = 100 volts / 10 min − 5 min.

Fig. 13 — Measurement of Power Absorbed by the Electric Field
Axes: Ordinate = ΔW (in watts); Abscissa = Voltage V.
Title: Additional power taken by charging wires when raising the sphere.
With CCl₄ (r = 6.8); K = 34/34 = 70%; ΔW_max = 20 watts; V_max = 1500.
Note: V × I is the voltage of the belt charging wires.

Fig. 14 — Sphere Voltage as a Function of Additional Power
Axes: Ordinate = V (in kilovolts), Sphere voltage; Abscissa = Additional power.
With CCl₄ (r = 6.8); K = 600/680 = 86.5%; Comparison with Wimshurst machine.

SECOND THESIS

PROPOSITIONS GIVEN BY THE FACULTY

On the absorption of ultrasonic waves in liquids.

Seen and approved
Paris,
The Dean of the Faculty of Sciences.

Seen and permission to print granted.
The Rector of the Academy of Paris.

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