GYROMAGNETIC EFFECTS AND ELECTRON INERTIA EFFECTS¹

S. Barnett, Los Angeles

Part I. Gyromagnetic Effects
Introduction. A. Preliminary, historical, and general remarks.
B. The macroscopic magnet as a gyroscope. Maxwell’s experiment.
C. Magnetization upon rotation (the Barnett effect). D. Rotation upon magnetization (the Einstein–de Haas effect). E. Gyroscopic magnetization in a rotating field.

Part II. The Effect of Electron Inertia
A. Introduction. B. Centrifugal experiments of Lebedev and Nichols. C. Ballistic experiments of Tolman and Stewart. D. Experiments of Tolman, Karrer, and Guernsey, and of Tolman and Mott-Smith. E. Barnett’s experiments on electron inertia.

Part I. Gyromagnetic Effects

Introduction

In this article two closely related groups of phenomena are considered: 1) magnetic or dynamic phenomena caused by the existence of elementary magnets that play the role of rotors or gyroscopes; these phenomena are known as gyromagnetic or magnetomechanical; 2) mechanical or electrical phenomena explained by the inertia of free electrons in conductors or of bound electrons in insulators. The gyromagnetic phenomena are considered in the first part of the present article, the others in the second.

A. Preliminary, Historical, and General Remarks

§ 1. The basic foundation of the phenomena under consideration

Every investigator who predicted the possibility of discovering one or another gyromagnetic phenomenon based his proof on the famous hypothesis of Ampère and Weber, according to which the magnetic element of any magnetic substance is an unchanging (long-existing) rapid

¹ Rev. Mod. Phys., 7, 129, 1937. Translation by N. N. Malov.

ing electric charge of molecular or intramolecular dimensions, possessing a known mass (or inertia). According to this hypothesis, the magnetic element must possess both a rotational (mechanical) and a magnetic moment, unless the element is formed by positive and negative electric charges rotating in opposite directions. Obviously, in this case the existence of only a magnetic moment is possible; in another case a definite mechanical moment may exist without a magnetic moment (when the directions of rotation of both charges are the same). In all other cases, however, the magnetic element must possess both the properties of a magnet and the properties of a gyroscope.

§ 2. A Simple Gyroscopic Model

In studying all gyromagnetic phenomena it is useful to make use of the gyroscopic model shown in Fig. 1; this model was first proposed by the author1. It differs from an ordinary gyroscope only by the presence of two additional cords \(SS\), made, for example, of rubber cord, and by a device for fastening them. A top resting on a ring can rotate rapidly about its axis \(A\). In addition to the action of the cords \(SS\), the ring and the axis \(A\) can move freely about the horizontal axis \(B\), and the angle formed by the axis \(A\) with the vertical axis we shall denote by the letter \(\theta\).

Fig. 1.

Further, the axis \(B\), together with the top and its holder, is capable of rotating about the vertical axis \(C\). If the top is set into rapid rotation about the axis \(A\) and at the same time the whole apparatus is slowly turned about the axis \(C\) (the centrifugal force may here be disregarded), then the top will rise, and the direction of its rotation will tend to coincide with the direction of the forced rotation about the axis \(C\), so that the angle \(\theta\) will thereby decrease. The greater the angular velocity of rotation about the axis \(C\), the more considerable the rise of the top. This rise would continue until the axes \(A\) and \(C\) coincided, if it were not hindered by the cords and by the mechanical resistances due to the inevitable imperfection of the mechanical construction of our model (we note once again that the influence of the centrifugal force has not been taken into account by us).

§ 3. Four gyromagnetic phenomena investigated up to the present time

In this section we shall consider only the qualitative side of these effects; a more detailed discussion of them is given below.

1. A macroscopic magnet as a gyroscope (Maxwell’s experiment, 1861; see also § 7). If all the magnetic elements of an iron (or other magnetic) body are identical and each possesses angular momentum, then the whole body, being magnetized in some direction, must acquire a certain (hidden) angular momentum with respect to this direction; therefore, if the body is set into additional rotation about another axis, it must behave like the gyroscope described in § 2: the body, as before, must tend to change its orientation in such a way that the direction of the (hidden) angular momentum approaches the direction of the forced rotation. This experiment, apparently the first of the gyromagnetic experiments, was performed by Maxwell in 1861, but gave negative results².

2. Magnetization by rotation (the Barnett effect, 1914, §§ 8—23). Maxwell did not have the idea of performing an experiment in which each of the enormous number of magnetic elements of a magnetic body would be oriented under the action of the body, and of measuring the macroscopic effect of the change in orientation of the elementary magnets by one of the magnetic methods.

The first experiment based on this idea was apparently carried out 40 years later by John Perry, who tried, but without success, to detect the magnetization of an iron rod when it was rotated. In 1909 the same idea occurred to the author of the present article, who at once, together with L. Barnett, began experiments that yielded positive results only in 1914, when they were published³. These were the first successful experiments in an entire series of other investigations of the gyromagnetic effect. They were published before all the others and were fully confirmed, both qualitatively and quantitatively, by a number of later investigations of the two effects considered and their inverses (see below). The qualitative classical theory of these experiments amounts to the following: if the body under investigation is set in rotation about some axis, then the magnetic elements, possessing angular momentum, tend to behave like the gyroscope described in § 2; all the magnetic elements tend to change their orientation in such a way that the direction of their rotation coincides as far as possible with the direction of the forced rotation. The coincidence would be complete if the influence of the remaining parts of the body under investigation on each of its magnetic elements did not make itself felt.

In the case of a simple ferromagnetic body in the ordinary state, only a weak change of orientation can be obtained; this is explained by the influence of neighboring elements of the body, corresponding to the role

...of the weights present in our gyroscopic model. Owing to the rotation, each magnetic element creates a small rotating and, consequently, magnetic moment parallel to the direction of the forced rotation; therefore a body whose magnetic elements, under ordinary conditions, are uniformly distributed in all directions, must, under forced rotation, become magnetized in the direction of the axis of the forced rotation.

If all the rotating electric charges of the magnetic elements are positive, then the body will become magnetized in the same direction in which it would be magnetized by an electric current flowing through a coil surrounding the body, in the direction coinciding with the direction of the angular velocity imparted to the body. If, however, all the charges are negative, or if negative charges predominate, a magnetization of the opposite direction must be produced. In reality, the latter is observed.

1. Rotation upon magnetization (the Einstein–de Haas effect, 1915–1916). If, as Maxwell supposed, an iron rod magnetized along its axis possesses a moment of rotation (hidden) with respect to this axis (this moment being the resultant moment of the individual magnetic elements), then any change in the magnetization must be accompanied by a change in this hidden moment, and therefore, according to the third law of dynamics, when the magnetization changes the rod must acquire a torque of the same magnitude but of opposite direction. This idea was expressed by O. Richardson4 in 1907. In the same year he gave a detailed theory of this effect and carried out a number of experiments which, however, gave no result. The first experiments that gave a satisfactory result both in the magnitude of the effect and in its sign were performed in 1915–1916 by Einstein and de Haas5, who, until the end of 1915, did not know of the successful investigation of the reverse effect by the author of these lines. Richardson’s work was likewise unknown to them (§§ 22–44).

2. Gyroscopic magnetization upon rotation of a magnetic field. [Experiments of Fischer6 (1922, 1924) and Barnett7 (1926, 1933).] In these experiments a rod (or toroid) made of magnetic material and, as far as possible, freed from residual magnetization was placed in a magnetic field perpendicular to the direction of the latter. This field (as well as the magnetization produced by it) was rapidly rotated; in this process the change in the longitudinal magnetization of the body was investigated. Such changes should not take place if only the magnetic elements do not participate in the rotation of the vector characterizing the intensity of the magnetization. But even this possible participation can scarcely be detected with the sensitivity of the instruments at our disposal at the present time.

In all these experiments a null effect was obtained (§ 45).

§ 4. Gyromagnetic Ratio

The most important quantitative characteristics of magnetic elements are their magnetic moment \(\mu_0\), their torque \(M_0\), and the ratio of the second to the first, called the gyromagnetic or magneto-mechanical ratio. This ratio, denoted by the letter \(\rho\), is defined by the equation

\[ \rho=\frac{M_0}{\mu_0} \tag{1} \]

As we shall see, the quantity \(\rho\) has been determined with great accuracy for a whole series of ferromagnetic and paramagnetic substances. It is the most important characteristic of all gyromagnetic experiments.

§ 5. Gyromagnetic Ratios for Various Magnetic Elements

1. Electron orbit (W. Weber, Rutherford, Bohr). Let us suppose that the magnetic element consists of one electron with mass \(m_0\) and charge \(e\), rotating in a circular orbit of radius \(r\) with constant angular velocity \(\omega\) (and areal velocity \(\alpha=\frac{1}{2}\omega r^2\)) around a considerably heavier nucleus with charge \(-e\); the nucleus may be regarded as practically immobile. In this case we have:

\[ \mu_0=e\alpha,\qquad M_0=m_0\omega r^2=2m_0\alpha \]

and

\[ \rho\left(=\frac{M_0}{\mu_0}\right)=\frac{2m_0}{e}=\rho_0 \tag{2} \]

If the orbit is elliptical rather than circular, then, as is easy to show, this quantity determines the ratio of the mean value of the rotational moment to the mean value of the magnetic moment.

2. Rotating charged body. Föppl investigated how magnetic elements consisting of homogeneous, uniformly charged rotating bodies behave in a magnetic field. He did not take into account that mass has an electromagnetic nature, and assumed that the mass density is everywhere proportional to the charge density. For such an element, as for an electron orbit, one obtains

\[ \rho=\frac{2m_0}{e}=\rho_0. \]

3. Rotating electron. M. Abraham\(^9\) investigated the behavior in a magnetic field of a rotating spherical electron, uniformly charged on the surface or throughout its volume; he calculated the moments under the assumption that the mass and the moment have a purely electromagnetic nature.

The masses \(m_s\) and \(m_v\) for an electron charged over the surface or throughout the volume turned out to be equal to

\[ m_s=\frac{2}{3}\frac{e^2}{a} \quad \text{and} \quad m_v=\frac{4}{5}\frac{e^2}{a}, \tag{4} \]

where \(e\) is the charge of the electron, \(a\) is its radius.

For the angular velocity \(\omega\), the corresponding moments of rotation are:

\[ M_s=\frac{1}{3}m_s a^2\omega \quad \text{and} \quad M_v=\frac{1}{7}m_v a^2\omega, \tag{5} \]

and the magnetic moments are expressed by the relations:

\[ \mu_s=\frac{1}{3}ea^2\omega \quad \text{and} \quad \mu_v=\frac{1}{5}ea^2\omega. \tag{6} \]

Hence we find the gyromagnetic ratios:

\[ \rho_s=\frac{m_s}{e} \quad \text{and} \quad \rho_v=\frac{5}{7}\frac{m_v}{e}=\frac{6}{5}\frac{m_s}{e}. \tag{7} \]

The first of these is one half as large as in the case of the electron orbit.

1. Ions and atoms with electron orbits, taking into account the spin of the electron. If the Landé splitting factor is denoted by \(g\), then an ion or atom considered as a magnetic element will be determined by the gyromagnetic ratio

\[ \rho=\frac{2m}{eg}=\frac{\rho_0}{g}. \tag{8} \]

From this expression we obtain:

\[ g=\frac{2m}{e\rho}=\frac{\rho_0}{\rho}. \tag{9} \]

Consequently, the splitting factor is numerically equal to the reciprocal of the gyromagnetic ratio expressed in fractions of the gyromagnetic ratio \(\rho_0\) for the electron orbit.

1. Complex elements. Complex elements consisting of nuclei and electrons have been considered by Richardson\(^{10}\) and others\(^{11}\).

B. The Macroscopic Magnet as a Gyroscope. Maxwell’s Experiment

§ 6. Relation between the magnetic moment and the hidden rotational moment of a macroscopic magnet

Let the magnet be magnetized symmetrically with respect to its axis, and let all its magnetic elements be identical. If \(\theta\) is the smallest angle between the axis of a magnetic element and the vector of the intensity of magnetization \(I\), then we have

\[ I=\sum \mu_0 \cos\theta, \tag{10} \]

where the summation is extended over a unit volume. The internal (hidden) rotational moment per unit volume is equal to

\[ j=\sum M_0\cos\theta=\rho\sum\mu_0\cos\theta=\rho I. \tag{11} \]

If \(I\) is the mean intensity of magnetization along the axis of the magnet, \(V\) its volume, then \(IV\) determines the magnetic moment, while the hidden rotational moment is determined by the expression \(M=\rho IV\). Thus \(\rho\) can be found from the values of \(M\) and \(IV\).

§ 7. Maxwell’s Experiment (§ 3)

We may now pass to the consideration of the quantitative theory of the first gyromagnetic experiment—Maxwell’s experiment. This experiment was carried out with apparatus somewhat resembling the gyroscope shown in Fig. 1. The top and the frame carrying it were replaced by a coil of wire carrying a current, or by an electromagnet, whose axis coincided with the axis \(A\) (Figs. 1 and 2), while the center of gravity lay on the axis \(B\). Such a body, if it possesses an internal rotational moment, must behave like the gyroscope considered in § 2. However, the centrifugal force cannot be neglected, especially since, in order to obtain a noticeable gyroscopic effect, fairly considerable velocities relative to the vertical axis are necessary. The influence of the centrifugal force can be reduced if the electromagnet is provided with additional loads placed along the axis \(CC\), normal to the axes \(A\) and \(B\), and properly centered.

Fig. 2.

Let \(A, B, C\) denote the moments of inertia of the system with respect to the axis of the magnet, the horizontal axis \(B\), and the axis \(CC\), normal to the first two. Let \(\theta\) be the angle between the axis \(A\) and the vertical direction (\(R\) in Fig. 2, \(C\) in Fig. 1), \(\Omega\) the forced angular velocity with respect to the vertical axis, \(J\) the total rotational moment of the system, \(M\) the hidden rotational moment, and \(\beta\) the angle between the moment \(J\) and the axis \(A\).

Suppose that, under the influence of weights producing a torque \(T\) in the direction of increasing the angle \(\theta\), the angular velocity \(\Omega\) and the angle \(\theta\) remain unchanged. The moment \(J\) may be resolved into two mutually perpendicular components: one of them, \(J\cos(\theta-\beta)\), parallel to the axis of forced rotation, remains constant. The other, \(J\sin(\theta-\beta)\), perpendicular to the first, changes with constant velocity

\[ T=\Omega J\sin(\theta-\beta). \tag{12} \]

But

\[ J\cos\beta=A\Omega\cos\theta+M \quad \text{and} \quad J\sin\beta=C\Omega\sin\theta, \tag{13} \]

therefore,

\[ T=(A-C)\Omega^2\sin\theta\cos\theta+M\Omega\sin\theta. \tag{14} \]

If \(C\) is somewhat greater than \(A\), then the torque \(T\) necessary for maintaining constancy of the motion can become zero, and the motion will be stable even when the supports are removed, provided that

\[ \cos\theta=\frac{M}{(C-A)\Omega}. \tag{15} \]

By means of two movable weights (on a screw), capable of being displaced along the axis \(C\), one can precisely choose the ratio between \(A\) and \(C\) (the axis \(B\) is the principal axis of the system) and make the apparatus very sensitive. Taking into account the disturbances introduced by the terrestrial magnetic field, one might expect that the results of the experiment would be rather rough. The experiment showed, however, that no changes in the value of the angle \(\theta\) upon changes of \(M\) and \(\Omega\) could be detected, even in those cases when an iron core was inserted into the coil.

Maxwell came to the conclusion that if a magnet, or a coil traversed by a current, does contain matter in concealed motion, then the moment of rotation created by this motion must be very small in comparison with quantities accessible to measurement.

Calculating \(M\) as the product of the constant \(\rho\), determined from other gyromagnetic experiments, by the magnetic moment of the magnet, and taking into account the fact that equation (14) is valid only in the case when the horizontal axis of rotation passes exactly through the center of gravity of the magnet, de Haas and de Haas–Lorentz\(^{12}\) showed that changes of \(\theta\) could scarcely be observed even under the most favorable conditions of a modern experiment.

If it were possible to make the difference \((A-C)\) very small and the angle \(\theta\) close to \(90^\circ\), then the quantity \(M\) could be determined from equation (14).

C. Magnetization upon rotation (Barnett effect)

§ 8. Theory of magnetization upon rotation (§ 3)

Let us suppose that a magnetic element is formed by a symmetric electrical system rotating with angular velocity \(\omega\), having magnetic moment \(\mu_0\) and rotational moment \(M_0=\rho\mu_0\); the rotation takes place about the axis of symmetry; all the rotating charges have one and the same sign\(^{13}\). The vectors \(M_0\) and \(\mu_0\) either coincide in direction or are mutually opposite, depending on whether the rotating charges are positive or negative.

Let \(A\) be the moment of inertia of the magnetic element with respect to

its axis of rotation, \(B=C\) is the mean moment of inertia with respect to any central axis normal to the axis of symmetry.

Let now a body, of which such elements are constituent parts, be set into rotation about the axis \(C\) with angular velocity \(\Omega\). Then the element, like the top of a gyroscope, tends to take such a position that its axis of rotation coincides with the axis of the forced rotation; but the influence of the remaining parts of the body, which create a certain additional moment \(T\), will hinder this rotation to a greater or lesser extent. After a short interval of time a steady state will be established, in which the axis of the magnetic element will continuously describe a cone, forming a constant angle \(\theta\) with a line passing through the center of the element parallel to the axis of the forced rotation \(C\). When this state is reached, \(T\) will be determined by equation (14), which may be written in the form

\[ T=\left[M_0\Omega+(A-C)\Omega^2\cos\theta\right]\sin\theta . \tag{16} \]

Let us now consider the same body, but instead of bringing it into forced rotation let us place it in a uniform magnetic field of intensity \(H\), directed along the former axis of rotation. Let us again consider a magnetic element, whose magnetic axis under the action of the field assumes a position in which it forms with the field \(H\) an angle \(\theta\). Under the action of the field the element tends to orient itself so that its axis coincides with \(H\), but this is hindered by the remaining parts of the body, which create an additional moment \(T'\). This moment is determined by the equation

\[ T'=\mu_0 H\sin\theta . \tag{17} \]

Let us now find such a field intensity as exerts on the orientation of the magnetic elements the same influence as the rotation of the body with angular velocity \(\Omega\). For this it is necessary to equate \(T\) and \(T'\). We find

\[ \mu_0 H\sin\theta=\left[M_0\Omega+(A-C)\Omega^2\cos\theta\right]\sin\theta \tag{18} \]

or

\[ H=\frac{M_0\Omega}{\mu_0}\left[1+\frac{(A-C)\Omega}{A\omega\cos\theta}\right], \tag{19} \]

The practically attainable values of \(\Omega\) are so small in comparison with the possible values of \(\omega\) that the last term may be neglected. Thus, for any magnetic element, independently of its orientation, we obtain with sufficient accuracy the following quantity:

\[ H=\frac{M_0\Omega}{\mu_0}=\rho\Omega=2\pi\rho\nu=\lambda\nu, \tag{20} \]

where \(\nu\) is the frequency of the forced rotation (revolutions per second).

and \(\lambda = 2\pi\rho\)—a quantity which in 1914 was called the “internal magnetic intensity of rotation.”

From what has been set forth above it follows that, if all the magnetic elements of a body are identical, then rotation of the body with frequency \(\nu\) produces the same magnetization as is obtained when the body is placed in a magnetic field of intensity \(2\pi\rho\nu=\lambda\nu\) oersted.

It is obvious that, under the action of a directing field, the magnetic element will precess uniformly.

If we denote by \(\delta H(=-H)\) the change in field intensity acting on the element during its precession, then from equation (8–4) one obtains

\[ \nu=\frac{H}{2\pi\rho}=-\frac{\delta H}{2\pi\rho}, \tag{21} \]

which coincides with the classical change of frequency in the Zeeman effect, arising in a field of intensity \(\delta H\), provided \(\rho=\rho_0\). Thus, under the action of the perturbing field \(\delta H\), the element performs Larmor precession with frequency \(\nu\), corresponding to the frequency of rotation of the body.

If in the body there are magnetic elements of two kinds, positive and negative, characterized by the constants \(\rho_1\) and \(\rho_2\), then rotation of the body produces the same effect as the action of a field \(H_1=\rho_1\Omega\) on the positive elements and of a field \(H_2=\rho_2\Omega\) on the negative elements.

If the influence of the negative elements is stronger, then the influence of rotation will create an additional magnetization in the direction \(H_2\), but its magnitude will be less than \(\rho_2\Omega\); the latter value would be obtained if the body contained only negative elements. In very weak fields all magnetic bodies acquire magnetic moments proportional to the intensity of the applied field. Similarly, since the values of \(\rho\Omega\), even at the greatest possible speeds, are equivalent to very small values of \(H\), these bodies, when rotated, must become magnetized in proportion to the speed of rotation.

If, however, one experiments with a ferromagnetic substance that is not in the demagnetized state or close to it, but in a state corresponding to a steeper part of the magnetization curve, then even a slight change in field intensity, or the use of small speeds, may prove sufficient to create significant, quite unmistakable changes in magnetization.

§ 9. Experiments with magnetization during rotation

Two types of experiments were used, substantially different from one another. The first successful experiment was carried out in 1914 (and repeated in 1915); in it a large iron rod about \(1\ \mathrm{m}\) long and \(7\ \mathrm{cm}\) ...

in diameter. The method was based on the phenomenon of electromagnetic induction2. The second method, in which considerably smaller rods of iron, cobalt, or nickel were used, is magnetometric3.

In both methods the rods were fastened along their horizontal axis and placed in a space where the earth’s field had been neutralized by special devices. The purpose of neutralizing the influence of the earth’s field is to eliminate the possibility of the formation of eddy currents in the rotating rod. Taking symmetry into account, one may, however, expect that the effect of these currents will be very small in the method investigating the phenomenon of electromagnetic induction. And indeed, the first successful experiment, carried out in the first half of 1914, was performed in the earth’s field. In the experiments, however, belonging to the second half of 1914 and to 1915, as well as in later work, the influence of the earth’s magnetic field was, as a rule, neutralized.

§ 10. Experiments by the method of electromagnetic induction

In work by the method of electromagnetic induction, the “internal magnetic intensity of rotation,” equal to \(2\pi\rho\nu\), was determined by comparing the change of the magnetic flux penetrating the rod, arising when the rod rotates about its axis with a known (measured in the experiment) speed, with the change caused by the creation of an additional uniform magnetic field of known intensity, directed parallel to the axis of the rod. The changes of flux are proportional to small intensities. These changes were determined ballistically by means of a fluxmeter, the coil surrounding the rod being included in the circuit of the fluxmeter. If \(D\) is the deflection of the fluxmeter when the direction of rotation of the rod is changed (rotation frequency \(\nu\)), and \(D_0\) is its deflection when the direction of the magnetic field of known intensity \(H_0\) is changed, then we obtain:

\[ \frac{2\pi\rho\nu}{H_0}=\frac{D}{D_0} \quad \text{or} \quad \rho=\frac{D}{D_0}\frac{H_0}{2\pi\nu}. \tag{22} \]

In the experiments two completely identical rods were used, arranged in parallel; their central parts were enclosed by identical, symmetrically arranged coils, as shown in Fig. 3. The coils were connected in series with each other and with the fluxmeter, and the direction of their turns was mutually opposite, so that any fluctuations of the intensity of the earth’s magnetic field, acting equally on both rods, could not produce a deflection of the fluxmeter needle. One of the rods—the compensator \(A\)—was immobile, while the other—the rotor \(B\)—was alternately rotated in opposite directions; the changes in its magnetization were determined at the moment of stopping

rod. In the calibration experiments, rods \(A\) and \(B\) were uniformly wound with insulated copper wire; in addition, in order to create a strictly uniform field, two wooden rods of the same diameter, provided with the same winding, were attached to the ends of the rotor. The earth’s field in the space occupied by the rotor was neutralized, in order to eliminate the occurrence of eddy currents and possible changes of the axial flux caused by a change in the shape or position of the rod, as well as by small oscillations of the rod axis during its rotation. In subsequent experiments the rod was rotated at the same speed in opposite directions, in order to eliminate the influence of changes in magnetization caused by expansion under the action of centrifugal force and by other distorting causes, including also heating of the rod supports. The experiments were also carried out with the rod axis turned through \(180^\circ\), which made it possible to eliminate the influence of a possible magnetic effect of twisting of the rod set in motion from one end.

Fig. 3.

§ 11. Results of experiments by the method of electromagnetic induction

Careful observance of the symmetry of the entire apparatus and the precautions taken to eliminate possible errors made it possible to bring the accuracy of the measurements to \(12\%\). It was found that the magnetization is proportional to the speed of rotation, as required by the theory. As for the sign of the rotating electric charges, in all cases it was found that, upon rotation, the iron is magnetized opposite to the direction in which it would have been magnetized if the magnetizing current flowed in the direction of rotation; hence it follows, as indicated above, that the rotating Ampère charges are negative.

The numerical results of the experiments of 1914 gave, for the gyromagnetic ratio, the value \(1.01\,\frac{m}{e}\). More accurate experiments of 1915, carried out by the same method with certain improvements, gave the value \(0.95\,\frac{m}{e}\). Taking into account the experimental errors, one may conclude that both results correspond only to one half of the value \(2\,\frac{m}{e}\) required by the theory proceeding from the conception of the electronic orbit as a magnetic element. In the calculations it was assumed that \(\frac{e}{m}=1.757\cdot 10^7\) CGSM.

These investigations gave the direct (and earliest) proof of the real existence of currents that had until then been hypothetical. It was proved that these currents are produced by negative charges possessing mass and inertia. In addition, an entirely new method of magnetizing bodies was found.

However, the magnetization obtained even at the most rapid rotations is very slight. Thus, as substitution of the value given above into the corresponding equation shows, rotating a body with a frequency of 100 Hz is equivalent to placing it in a magnetic field amounting to only hundred-thousandths of the Earth’s magnetic field.

§ 12. Gyromagnetic anomaly and the nature of the magnetic element

One of the most important results of these experiments was the obtaining of a numerical value for the gyromagnetic ratio, which, as was found, is only half the value calculated on the assumption that the electron rotates in an orbit. This disagreement between theory and experiment received the name of the gyromagnetic (or magnetomechanical) anomaly. This result shows (with great probability), when compared with the data of § 5, that the magnetic element in iron is produced by the Lorentz electron owing to its rotation about its own diameter, and not owing to its motion in an orbit.

More precise results, obtained by the author together with L. Barnett at a later time, showed that the orbits also participate to a certain extent in the formation of the magnetic element, so that the gyromagnetic ratio for ferromagnetic substances is found to be somewhat larger than \(\frac{m}{e}\) (see below §§ 22, 23, 41).

§ 13. Experiments by the magnetometric method. Arrangement of the apparatus

Later experiments carried out by the magnetometric method were first published in 1917; they dealt chiefly with iron, cobalt, and nickel; a more extensive and precise series of measurements was performed in 1923–1924.

In the magnetometric method an astatic magnetometer was arranged in such a way that the center of its lower magnetic system lay on the axis or in the equatorial plane of the rod under investigation. The deflections of the magnetometer, arising when the direction of rotation of the rotor, whose speed was known, was changed, were compared with the deflections obtained when the direction of a known magnetic field parallel to the axis of the rotor was changed. These deflections in both cases are proportional to the change in the magnetic moments, which in turn are propor-

proportional to the internal stress under rotation or to a previously programmed stress of the field. Obviously, equation (10-1) is applicable to this case.

The elimination of possible errors in the magnetometric method is more difficult than in the preceding one, but it has the considerable advantage that its sensitivity is substantially greater and it does not require such large specimens under investigation as the previous method.

Figure 4 shows a diagram of the arrangement of the parts of the instrument. A light vertical aluminum rod \(I\) carries two systems of very small horizontal magnets \(F\) and \(J\), forming a precisely adjusted astatic system. The rod \(I\), together with the magnets, is suspended on a thin thread \(B\) of fused quartz from the torsion head \(A\). Further, on the rod there is a damping metal vane \(D\), located between two parallel plates not shown in the drawing, and a small mirror \(E\), which makes it possible to measure the deflections of the magnetometer by the ordinary method of a tube and scale. To eliminate the influence of air currents, the entire suspended system is enclosed in a metal casing provided with glass windows and wrapped in cotton wool or another heat-insulating material. In the casing there is a small amount of radioactive salt, which prevents electrification of the moving parts.

Fig. 4.

\(H\) represents two small control magnets that produce a field directed either parallel or perpendicular to the axis of the upper magnets of the astatic system. These magnets make it possible to adjust the zero point of the system and to change the sensitivity of the instrument independently of one another \(^{16}\). In each part of Fig. 4 only one of two mutually perpendicular magnets is visible. Without this adjusting system, used for the present work and for other precise investigations, or without an equivalent system of coils carrying an electric current, reliable measurements would have been almost or even entirely impossible.

The rod under investigation—the rotor—was arranged so that its axis was directed from east to west; it was placed near the lower magnetic system of the magnetometer. A rod \(C\) completely similar to it—the compensator—was placed in a parallel horizontal plane, as far as possible at the same distance from the opposite pole of the upper magnetic system, but somewhat

farther north or south. By means of slight displacements of this compensator it was possible easily to compensate the effect on the rotor and on the magnetometer of continuous fluctuations in the intensity of the earth’s field. The device described made it possible to avoid a whole series of accidental measurement errors. However, despite all these precautions, the most critical measurements were carried out late

Fig. 5.

at night (the same applies to experiments by the method of electromagnetic induction), when the disturbing influence of the sun on the earth’s magnetic field is considerably less than in the daytime; usually the measurements were made after two o’clock in the morning, when interference from the magnetic fields produced by streetcars was minimal (by fortunate chance, the most critical part of the work was carried out during a minimum of sunspots). In calibrating the instrument a small Helmholtz coil was used (placed near the lower magnets at the initial stage of the work and near the upper ones in the later measurements) and a solenoid \(S\), which was slipped onto the rotor.

In Fig. 5 a schematic representation is given of the entire installation (not entirely corresponding to its final form).

On the frame \(RR\) are located coils that neutralize a large part of the earth’s magnetic field. The coils \(TT\), mounted near the upper magnets and traversed by the same current as the coils of the frame \(RR\),

provide almost complete independence of the zero point and sensitivity of the magnetometer from the current strength. The rotor together with its supports is denoted by the letter \(U\).

In the measurements the lower magnets of the magnetometer were placed either on the axis of the rotor (axial position), or else near the equatorial plane passing through the center of the rotor (equatorial position). In the later works the magnets were always arranged in a vertical plane perpendicular to the axis of the rotor. Their arrangement for both positions is shown in Fig. 6 (\(B\) and \(A\)).

Fig. 6.

All measurements by the magnetometer method, except the very earliest ones, were carried out in a small room built of nonmagnetic materials, in which the earth’s magnetic field was almost exactly homogeneous.

§ 14. Calibration process. Slow rotations

The calibration method was essentially reduced to three processes (1), (2), and (3), for which three corresponding standard coils were required.

In process (3), for convenience and accuracy it was necessary that the rotor be in uniform rotation, which made it possible to observe the mean effect. If the earth’s magnetic field is exactly compensated, this necessity is due to the fact that the field around the magnets of the magnetometer, created by the residual magnetization of the rotor, depends on the position of the rotor both in magnitude and in direction. If, however, the earth’s field is not completely compensated, then an additional influence on the magnetometer arises, caused by the fact that the magnetic susceptibility of the rotor is not entirely symmetric with respect to its axis.

In processes (2) and (1), when only the ratio of sensitivities was investigated, with slow rotations it is necessary to take into account

only the defects of the rotor with respect to axial symmetry. Even with an asymmetric rotor, in process (1) no noticeable error is obtained, and in process (2) only a small error.

In almost all the experiments described here the rotor (in processes 2 and 3) was rotated uniformly and slowly, but nevertheless at a speed sufficient to obtain distinct deflections of the magnetometer mirror. In some particular cases the calibration was carried out at speeds corresponding to the speeds of the principal experiments.

§ 15. Calibration

(A). Process 3. Absolute sensitivity of the magnetometer. In process 3 the sensitivity of the magnetometer itself was determined, almost independently of the features of the rotor and compensator; for this purpose the direction of a small current of known magnitude \(J\) was changed in a small tertiary standard coil \(C\), placed at the upper or lower magnets; by moving the control magnet the desired sensitivity was set, i.e. the appropriate deflection \(c\) of the magnetometer. This quantity represents the absolute sensitivity of the magnetometer. In this process, as in processes (1) and (2), observations were made at strictly definite times, and current reversals were made at equal intervals.

(B). Process 2. In this process the ratio of the sensitivity of the rotor to the absolute sensitivity was determined. The secondary standard solenoid \(B\) (\(S\) in Fig. 4) was slipped over the rotor and held in a coaxial position by the corresponding stand. Through the solenoid the same current was passed as through coil \(C\) in process 3, and the deflection of the magnetometer upon changing the direction of this current was determined. This quantity determines the sensitivity of the rotor \(b\).

The ratio \(Q=\dfrac{b}{c}\) was determined for the given rotor with a given position of the magnetometer system. In precise measurements, especially in those cases when the coils \(C\) were located at the upper magnets, it is necessary that the earth’s field be compensated, as in the principal experiment, because \(Q\) depends on the ratio of the moments of both magnetic systems of the magnetometer, and this ratio is different with compensation and without it; the difference reaches \(\frac{1}{3}\%\).

(C). Process 1. Reduction to a uniform field. If the calibrating solenoid \(B\) (the secondary standard coil), which is only slightly longer than the rod, had infinite length, then in the equatorial position of the magnetometer no effect should have been observed; in that case the rotor would be in an ideally uniform axial field, similar to the field arising in it during rotation. It is neces-

the need to shorten solenoid \(B\), connected with the requirement that the magnetometer be installable in the axial position, compels one to carry out one more calibration process (1), in which, for each rotor and each position of the magnetometer, the value of the rotor sensitivity \(a\) is determined that would be obtained with an infinitely long solenoid and zero influence on the magnetometer in the equatorial position. This is achieved with the aid of the first standard solenoid \(A\), entirely similar to solenoid \(B\), but considerably longer. The ratio \(\dfrac{b-a}{b}\) is hereafter denoted by the letter \(K\). For the rotors used in the main part of the work, \(K\) ranges from \(0.0\%\) (permalloy) to \(5.1\%\) (Fe—Ni alloy of Hopkinson).

§ 16. Remarks on the observations in the main experiments

In order to reduce random errors caused by fluctuations of the earth’s field and other causes, it was of course necessary, in the main measurements, to make a large number of readings, taking them at definite moments of time, at equal intervals, as when carrying out processes (1), (2), and (3). To eliminate various systematic errors (§§ 19, 20), it was necessary to rotate the rotor at constant speed alternately in opposite directions. In the last part of the work the observations were made in series of 12 observations with intervals of 30 sec. The results obtained were processed in the usual manner.

§ 17. Fundamental equation for the experimental determination of the quantities \(\lambda\) and \(\rho\)

Let \(\lambda \nu = 2\pi \rho \nu\) represent the internal intensity of the magnetic field when the rotor is rotated with frequency \(\nu\) Hz. Let \(d\) be the deflection of the magnetometer obtained when the direction of rotation is reversed. Further, let \(H_0\) be the intensity of the homogeneous magnetic field which would be produced in the space occupied by the rotor, when a current \(I\) flows through solenoid \(B\), assumed infinitely long; let \(a\) be the deflection of the magnetometer obtained when the direction of the current is changed in this case, and \(b\) the deflection obtained when the direction of current \(I\) is changed in the actually existing solenoid \(B\), mounted coaxially on the rotor. Then from equation (22) and § 15 we obtain:

\[ \frac{\lambda \nu}{H_0}=\frac{d}{a}=\frac{d}{b(1-K)} \tag{23} \]

or

\[ \lambda(=2\pi\rho)=\frac{dH_0}{b(1-K)\nu}\ \text{gauss/rev. per sec.} \tag{24} \]

\(H_0\) is determined by the product \(4\pi nI\), where \(I\) is the current strength,

\(n\) is the number of turns of the solenoid per unit length, \(b\) is determined from the sensitivity of the magnetometer \(C\) and the ratio \(Q=\dfrac{b}{c}\). The frequency \(\nu\) is determined from the rotation frequency \(N\) of the motor and the equation \(\nu=gN\), where \(g\) is the transmission ratio from the motor shaft to the rotor axis. Therefore we obtain:

\[ \rho\left(=\frac{\lambda}{2\pi}\right)=\frac{2nld}{Qc(1-K)GN}\ \text{gauss/radian. in sec.} \tag{25} \]

or

\[ \frac{\rho}{m/e}=\frac{2nld}{Qc(1-K)GN}\cdot\frac{e}{m}. \tag{26} \]

The results obtained by this method are in good agreement with the results computed from the standard observations by means of equation (26), see Table 2.

§ 18. Equation for determining \(\rho\) by comparison with the value \(\rho_s\) for the standard rotor \(s\)

To check the results obtained by means of equation (26), the following method was also used:

1. The value \(\rho_s\) was determined from direct observations with the standard rotor (steel III).

2. From four (in a few cases—three) separate coordinated series of observations, the quantity \(R=\dfrac{Q_i}{Q_s}\) was determined, i.e. the ratio of the value of \(Q\) for each rotor under investigation to the value \(Q_s\) obtained for the steel-III rotor.

3. The value of \(\rho\) for each rotor under investigation was determined from the equation

\[ \rho=\frac{\rho_s Q_s}{Q}=\frac{\rho_s}{R}. \tag{27} \]

§ 19. Systematic errors. Class A errors

(independent of the magnetization of the rotor)

In the course of the work it was necessary to investigate and eliminate a whole series of sources of systematic errors. Apart from errors in the standards, which were so small that they could be neglected, all the most important systematic errors subject to investigation and elimination could be divided into two classes, namely: class A—errors independent of the magnetization of the rotor, and class B—errors dependent on the magnetization of the rotor. Let us briefly consider the principal of these errors, beginning with class A errors.

1. Eddy currents in the rotor caused by incomplete compensation of the homogeneous (a) or inhomogeneous-

north–south (b) component of the magnetic field. These errors were not taken into account in the work by the method of electromagnetic induction, since the apparatus was very symmetrical and the accuracy of the observations was relatively low. In the work with the magnetometer these errors were more important, and the study and elimination of them required a great expenditure of labor and time. The influence of eddy currents was reduced to a negligibly small value by the most accurate possible adjustment of the strength of the direct current in the coil neutralizing the earth’s field; by setting the magnets of the magnetometer in such a position that the influence exerted on them by the eddy currents was minimal; by using magnets and coils with a very small moment; and, finally, by creating a sufficient distance between the rotor and the compensator (with small moment), and between the rotor and the control magnets. Taking into account the existence of continual fluctuations of the earth’s magnetic field, in order to ensure the setting of the proper current strength in the coils compensating the field it was necessary to carry out an enormous number of control measurements and to set up three variometers for measuring changes in the earth’s field in the north–south and east–west directions, as well as its vertical component. The influence of the residual field was made negligible (or very small), which was checked by special experiments, chiefly by means of the rotation of a copper rod.

1. Electric currents caused by the thermal effect in the supports. A noticeable effect was observed only with the copper rotor. In the final part of the work it was eliminated by sufficiently removing all heated supports from the magnetometer.

2. Electric currents caused by the thermal effect produced by the motion of air around the frame of the instrument during rotation of the rotor. This effect was not observed, but in the final part of the work its possible influence was eliminated by shielding the rotor from the frame with the aid of a cardboard tube fitted coaxially to the rotor.

3. Eddy currents and other electric and magnetic phenomena in the rotor and the rest of the apparatus. In the final part of the work this phenomenon was made insignificant by carefully neutralizing the magnetic field in the space where the moving parts were located, by using a collectorless alternating-current motor, and by choosing a sufficiently large distance to the magnetometer.

4. Thermal influence on the magnetometer caused by air currents flowing around the magnetometer during rotation of the motor. This effect, the existence of which had been assumed from the very beginning of the experiments, could be considerable; moreover, it manifested itself in the equatorial position considerably more strongly than in the axial one. Careful wrapping of the greater part of the magnetometer box with cotton wool and paper (the latter was required for other reasons) made it possible to eliminate this effect even in the absence of the cardboard tube mentioned above.

1. Vibrations of the magnetometer casing, affecting the suspension system differently for two different directions of rotation. Everything possible was done to reduce the oscillations as much as possible: the rotor was balanced insofar as possible; the same was done with all moving parts; the rotor frame, its motor, and the gearbox were carefully secured; the motor and the transmission from it to the rotor were removed from the magnetometer. The magnetometer stand was made as massive as possible. Even without this last precaution, conditions were obtained under which, with a well-demagnetized rotor, the effect of vibrations did not appear, except in certain isolated cases when the motor mountings accidentally became loose.

§ 20. Class B errors (depending on the magnetization of the rotor)

The following sources of errors depending on the magnetization of the rotor may be indicated (if it had been possible to demagnetize it completely, these errors would have been eliminated).

1. Twisting of the rotor, which was set in motion from one end, while the other end slid by friction in the support. The effect of twisting could be neglected when working by the method of electromagnetic induction, when a large steel rod was used, but it often appeared in the magnetometric method, whose accuracy was considerably higher.

These errors were of two types, depending on whether residual or induced magnetization was present. The former disappeared when the residual magnetization was destroyed, the latter when the magnetic field was destroyed. For very small torsions the former were proportional to the torque and changed sign when the direction of rotation changed. The latter errors are proportional to the square of the torque and do not depend on the direction of rotation; their influence was not observed in this work. If both shaft journals and both bearings are perfectly identical, the rotor is balanced and its magnetization is constant, and if the friction of the shaft journal does not depend on the direction of rotation, then the first effect can be eliminated by making measurements first with one direction of the magnetic axis of the rod, and then after turning it through 180° by shifting the rotor in its supports. Much effort was spent on making the shaft journals and bearings as identical as possible; in the most recent work the diameter of the shaft journal was equal to \( \frac{1}{8} \) or \( \frac{3}{16} \) inch, and the bearings were made of agate. They were lubricated with oil used for lubricating watches. Since the friction of the shaft journal was almost independent of speed, the final error at high speeds was considerably smaller than at low speeds.

1. Thermal effect of the rotor, caused by friction of the shaft journals. This effect introduced a systematic error,

since the magnitude of the heating varied with the change in the direction of rotation. When the conditions indicated in item 1 were fulfilled, the error due to heating, like the error due to twisting of the rotor, was eliminated by shifting the rotor in its supports. It was reduced as far as possible by fulfilling the conditions indicated in item 1 and by separating the shaft necks from the magnetic part of the rotor with gaskets made of a material having good heat-insulating properties.

1. Thermal effect of the air entrained by the rotor. This error may arise owing to air currents produced during the motion of the rotor. As a result of incomplete symmetry, different heating of the rotor is possible for different directions of its rotation. This influence was reduced by using a cardboard tube coaxial with the rotor, described in § 19. The error was eliminated by reversing the rotor.

2. Error due to centrifugal expansion of the rotor and other deformations (apart from twisting) arising during rotation. These errors were eliminated in observations at constant speed and opposite directions of rotation. If a systematic difference was observed upon changing the direction of rotation, then the error due to residual magnetization could be eliminated by reversing the rotor. But if the error was caused by axial induced magnetization, it cannot be eliminated in this way. It was shown, however, that even when magnetization considerably exceeding the magnetization that could have occurred in the experiments was produced, no influence on the magnetometer was observed.

3. Error due to axial displacement of the rotor. During rotation of the rotor there is always some axial displacement of it; if this displacement depends on the direction of rotation of the rotor, it may introduce a systematic error. This may occur for three reasons: 1) eddy currents caused by the presence of a residual field will exert different influences on the magnetometer depending on the direction of rotation; 2) the influence exerted on the magnetometer by the residual magnetization of the rotor will be different for different directions of rotation; 3) if the residual axial field has a gradient, then the induction in the rotor, as well as its influence on the magnetometer, will depend on the direction of rotation. All these effects were eliminated in the measurements described. The influence of effect 2, if it manifested itself, was eliminated by reversing the rotor.

4. Error due to changes in the azimuth of the rotor. If the (very small) angle between the normal to the rotor axis and the component \(\Delta x\) of the uncompensated magnetic field lying in the magnetic meridian (practically perpendicular to the rotor axis) changed, when the direction of rotation was reversed, by an amount \(\Delta a\), then the influence of this was equivalent to a change in the intensity of the axial field in the rotor by the amount \(f = \frac{1}{2}\Delta a \Delta x\); but

the mean results for all speeds agree well with one another.

As was indicated above, the theory requires that for each rotor the deflection of the magnetometer be proportional to the speed. In Fig. 11 the experimentally obtained dependence is shown between the speed and the reduced deflection of the magnetometer according to the data of series B. By reduced deflection (for each rotor) is meant the actual deflection, reduced to a constant sensitivity of the magnetometer, divided by the change in moment obtained at a constant small intensity of the magnetic field.

Fig. 11.

The dependence obtained is exactly linear.

§ 23. Results for the individual rotors from series B. Final mean value

Table 2 gives the observational data at the maximum speed, which, as indicated, is the most favorable for measurements; the results for smaller speeds differ little from the results given. The values of \(\rho \dfrac{e}{m}\) in the third column are calculated directly from the observations of series B by formula (26). The values placed in the fourth column were calculated by the method set forth in § 18, with the aid of formula (27), the value of \(\rho \dfrac{e}{m}\) for the material steel III, obtained in series B with transmission ratio \(\dfrac{2}{1}\), being taken as the standard. In the fifth column the mean of these two values is given.

The discrepancies between the data of the third and fourth columns have no systematic character; the mean of the numbers in the fourth column exceeds the mean of the numbers in the third column by only 0.002; the mean difference without regard to sign is \(\pm 0.015\).

Here a slight doubt may arise as to the reliability of the mean values given in Table 1, which agree well for two series of observations at two different speeds. However, it would be imprudent to conclude from the data of Table 2 that the values of \(\rho\) for different materials differ substantially from one another, since the discrepancies in the values of \(\rho\) for different rotors made of one and the same material approximately correspond

Table 2

\[ \rho\,\frac{e}{m}\ \text{for various rotors} \]

Series B. Speed 61 rev/sec, \(\dfrac{e}{m}=1.757\cdot 10^7\) CGSM

Rotor	Observation series number	\(\rho\cdot\dfrac{e}{m}\)	\(\rho_Q\cdot\dfrac{e}{m}\)	Mean of columns 3 and 4
Electrolytic iron II	18	1.067	1.092	1.080
Armco iron	10	1.026	1.022	1.024
Norwegian iron	6	1.032	1.052	1.042
Steel III	15	1.050	(1.050)	(1.050)
Steel IV	21	1.054	1.049	1.052
Steel I	9	1.049	1.044	1.046
Nickel I	9	1.049	1.020	1.034
Nickel III	10	1.014	1.003	1.008
Cobalt II	13	1.073	1.109	1.091
Heusler alloy I	10	1.012	1.031	1.022
Permalloy	16	1.057	1.036	1.046
Iron-nickel	10	1.015	1.016	1.016
Iron-cobalt	6	1.071	1.060	1.066
Cobalt-nickel	4	1.070	1.077	1.074

discrepancies obtained in the investigation of rotors made of different materials. Nevertheless, the mean values and a number of individual values agree well with the results obtained in studying the inverse effect (Table 4), which clearly indicates the existence of differences in the values of \(\rho\) for some materials.

It is very probable that the mean value of \(\rho\dfrac{e}{m}\), obtained in series B at the maximum speed, is equal to 1.051, and the error in its determination does not exceed \(2\%\).

D. Rotation on magnetization (Einstein–de Haas effect)

§ 24. Rotation on magnetization. General remarks on the experimental methods and theory

In all experiments devoted to the study of this effect, a circular cylinder of the substance under investigation was used; it was suspended vertically by its axis in a fixed frame by means of a vertical wire or thread stretched along the axis, or by means of two wires or threads, one of which was stretched from above and the other from below. The cylinder was placed in a coaxial magnetizing coil of insulated wire, attached to the cylinder or to the frame. The motions of the cylinder arising when its axial magnetization was changed were studied.

If the axial magnetic moment \(\mu\) changes by an amount \(\mu\), then the hidden angular momentum \(M=\rho\mu\) changes by the amount \(\rho\mu\). The angular momentum \(J\) of the rotor then changes by the amount

\[ g=\dot J=-\dot M=-\rho\dot\mu, \tag{28} \]

which represents the gyromagnetic moment acting on the rotor and magnetizing the coil. In most works it is assumed that the moment acting on the coil is vanishingly small. The frictional force, according to experience, is proportional to the angular velocity of the rotor.

If there are parasitic torques causing a change in \(J\), then, of course, equation (28) loses its validity. We shall assume at first that all such moments, which can be eliminated or taken into account by suitable methods, are completely absent.

For the investigation of this effect two methods have been proposed—the ballistic and the resonance methods. The resonance method has two modifications: a simple one (with one moment) and a complex one (with several moments).

In most works carried out by the resonance method, a rectangular or strongly smoothed (almost rectangular) form of the current curve was used; therefore the first harmonic of the magnetization was in phase, or almost in phase, with the first harmonic of the current, so that determining its amplitude presented no difficulty. In some works, including the first work of Einstein and de Haas, this condition was fulfilled by applying large amplitudes of magnetic-field intensity, for which the state of saturation was reached already at the beginning of each half-period. In the remaining works, to obtain a smoothed curve in weak alternating fields, a battery with a commutator, or something similar, was used.

§ 25. Ballistic method (Richardson, Stewart, Chattock and Bates)

In the ballistic method, equation (28) is applied in integral form:

\[ \delta J=\int g\,dt=-\rho\,\delta\mu, \tag{29} \]

it being assumed that the rotor is initially at rest. Thus here one determines the change in the angular momentum of the rotor \(\delta J\) under changes of its magnetic moment \(\delta\mu\), produced by a change of the current in the coil, or by some other means.

The value \((-\rho)\) is determined as \(\dfrac{(\delta J=J)}{\delta\mu}\); \(J\) is determined from the known formula

\[ J=(AK)^{\frac12}\theta, \tag{30} \]

where \(\theta\) is the angle of twist which, in the absence of damping, would be obtained upon the creation of the magnetic moment \(\delta\mu\), \(K\) is the moment of inertia of the system, and \(A\) is the torsional constant. Starting from the usual theory, it is easy to show that \(\theta\) can be determined from the actual angle \(\theta_0\) by the equation:

\[ \theta=\left(\frac{d_1}{d_2}\right)^{\frac{1}{\pi}\operatorname{arc\,tg}\frac{\pi}{\lg\frac{d_1}{d_2}}}, \tag{31} \]

where \(\frac{d_1}{d_2}\) is the ratio of two successive deflections in opposite directions. One may also write

\[ \theta=\theta_0(1+\lambda), \tag{32} \]

where \(\lambda\) is equal to one half of the logarithmic decrement of damping, if the damping is sufficiently small. Sometimes this formula proves inapplicable, which can possibly be explained by the inertia of the air surrounding the rotor and carried along by it during its motion (§ 39).

Wishing, as far as possible, to reduce the disturbing influence of the field of the magnetizing coil on the rotor, Stewart stopped the galvanometer when the field disappeared, but the residual magnetic moment of the rotor was sufficiently large, and then observed the deflection upon the creation of a weak field of the opposite direction, destroying the magnetic moment.

Chattock and Bates measured the deflection obtained upon reversal of the residual moment. In essence this idea belongs to Einstein\({}^{17}\).

In Stewart’s observations systematic measurements were made for both directions of residual magnetization; in the work of Chattock and Bates the reversal of the residual magnetization in both directions was likewise systematically investigated. In this way they tried to separate the influence of magnetostriction (§ 37) from the mean effect produced by the constant component of the vertical magnetic moment of the specimen, while the uncompensated magnetization produced by the vertical component of the earth’s field was considered vanishingly small in comparison with the magnetization produced by the coil.

Since the magnetostrictive effect could either add to the gyromagnetic effect or be subtracted from it, the error introduced by magnetostriction had to be excluded from the mean determined from many observations in which the rotor was repeatedly oriented in the same way.

In the observations of Chattock—Bates, systematic observations were made for two azimuths of the rotor differing by \(180^\circ\). This method made it possible to eliminate the systematic error caused by the influence of the uncompensated part of the horizontal component of the earth’s field.

§ 26. Simple resonance method (Einstein and de Haas).

In this method the only moment acting on the system is the gyromagnetic moment \(g=-\rho\mu\).

Method 1. A system possessing a natural frequency \(\nu_0\) is magnetized by an alternating current whose first harmonic has a constant amplitude and a frequency \(\nu\), which can be varied within narrow limits on both sides of \(\nu_0\). The relation between the frequency \(\nu\) and the half-angle \(\alpha\) of the vibrations of the system is determined. Knowing \(\alpha_0\) (the value of \(\alpha\) at resonance), the moment of inertia \(K\), the logarithmic decrement \(\lambda\), and the frequency \(\nu_0\), one can calculate the amplitude of the rotating moment at resonance \(J\) from the formula

\[ J=\pi\lambda K\nu_0\alpha_0. \tag{33} \]

Knowing \(J=\delta J\) and \(\mu=\delta\mu\), the amplitude of the first harmonic of the magnetic moment, one can calculate \(\rho\) by means of equation (29).

Method 2. If one measures a series of values of \(\alpha\) on the resonance curve (at different frequencies \(\nu\)), then one can determine \(\alpha_0\) and eliminate \(\lambda\), so that measuring it becomes unnecessary.

If the frequency is very low, as was the case in the later works of Einstein and de Haas, then the phase difference between the current and the displacement of the rotor can be determined by direct visual observations, whereby not only the magnitude \(\rho\) is found, but also its sign. At high frequency this is impossible, and one has to use some oscillographic scheme, as was done in the work of Beck and in the first work of Einstein and de Haas.

§ 27. Compound resonance method (de Haas[^18], Chattock, Sucksmith and Bates[^19], Barnett[^20]).

By adding to the gyromagnetic moment one or several additional moments varying synchronously with the gyromagnetic one, it is possible to achieve cancellation or reversal of the gyromagnetic effect, or else to obtain a moment in quadrature with it, of the same or opposite phase.

The first attempt to make use of this method belongs to de Haas, who (in 1916) attached to an iron vibrating system—a small investigated rod—a small permanent magnet, whose axis was horizontal. A small fixed coil, the turns of which were parallel to the magnet and whose axis passed through the magnet, was connected in series with the magnetizing coil and was supplied with a shunt that made it possible to regulate the intensity of the field it produced. The moment produced by the coil, of course, had to vary also when its distance from the magnet was changed. The supply was by a current with a smoothed curve form; the coil produced a moment in quadrature with the gyromagnetic moment and was intended to neutralize the influence of other moments that were in quadrature.

In a similar work by the author, the coil and magnet were used in an analogous manner, but the coil was arranged so that the magnet was at its center; it was connected (through a large regulating resistance) directly to the ends of the commutator magnetizing the circuit, or to a variable noninductive resistance connected in series with the magnetizing circuit.

If the external coil acting on the permanent magnet is connected, through a suitable resistance, in series with a solenoid or another coil surrounding the rotor, then, as is shown below, the torque produced by it will be in phase with the gyromagnetic torque, or will be opposite to it in phase, depending on the method of connection. The total torque (with the exception of torques in quadrature) can be made equal to zero, so that a null method of measurement is obtained. This idea was expressed by Chattock and first realized in his laboratory by Sucksmith and Bates. It was also used by the author, sometimes in connection with the coil mentioned above, producing an additional torque in quadrature.

Fig. 12.

Labels in the figure:

• Generator or battery
• Armature
• Platinum rod
• Silver wire
• Induction solenoid (resistance \(Y\))
• Rotor and magnetizing coil
• Induction circuit (conductance \(X\))
• Regulating resistance
• Mirrors
• Coil producing the additional torque (resistance \(b\))
• Platinum rod
• Permanent magnet (moment \(m_0\))
• Silver wire
• Tensioning weight
• Oil
• \(-\rho = b\gamma_0 m_0 k_0\)

In the null method, to the lower end of the rotor \(F\) (Fig. 12) there was attached a vertical nonmagnetic rod \(J\), which carried two small parallel mirrors \(I\), turned in opposite directions; below them was placed a permanent magnet \(L\) (with moment \(m_0\)), whose axis was horizontal. Still lower, on the suspension, was placed a heavy brass or copper weight \(N\), lowered into a vessel with oil or another damping liquid. To eliminate certain systematic errors (§ 31 and the following), observations were made in two mutually opposite azimuths, for which purpose the torsion head of the apparatus and the entire suspended system were turned through \(180^\circ\). The apparatus was surrounded by a coil which, when a suitable current was passed through it, neutralized the action of the earth’s field (in the author’s experiments all components of the field were neutralized) in that part of space where the rotor was located. The rotor was excited at the resonant frequency \(\nu_0\), and a gyromagnetic torque arose. By means of a small coil \(K\) (the main coil, producing a torque, being the constant \(\Gamma\)), surrounding the permanent magnet

(axis of the coil perpendicular to the magnet) and traversed by a current \(i\) of frequency \(\nu_0\), one could create an external torque \(c\) acting on the vibrating system. The magnitude of the torque was regulated by changing the current intensity \(i\).

In the null method, a torque \(c\) was produced equal in magnitude and opposite in phase to the torque \(g\), as a result of which the oscillations were annihilated, provided there were no extraneous torques.

In the simplest arrangement used by the author, the coil surrounding the rotor was a long coaxial solenoid \(E\), which we shall call the induction solenoid; its constant \(\gamma\) was almost constant throughout the entire space occupied by the rotor. In the work of Sucksmith and Bates a solenoid was used whose length was equal to the length of the rotor; in this case it was necessary to calculate a correction for the nonuniformity of the field for each rotor. The induction solenoid was connected in series with the coil \(K\) and the resistance box \(H\); we shall denote the total conductance of the circuit by \(X\). With the appropriate connection of the coils (depending on the sign of \(\rho\)) and a suitable value \(X=X_0\), the amplitude of the oscillations (if there are no external influences on the system) should, as is proved in the following paragraph, become zero.

In almost all of the author’s works and in de Haas’s work the magnetizing coil was placed directly on the rotor \(F\) (Fig. 12). In the author’s later works the rotor was magnetized by a long immobile coaxial solenoid placed inside the induction solenoid. In the work of Sucksmith and Bates the magnetizing coil was placed outside the induction solenoid and was coaxial with the rotor.

§ 28. Theory of the Combined Null Method

Let \(\varphi\) be the magnetic flux produced through the induction circuit when a magnetic moment \(\mu\) arises (the moment produced by the magnetizing coil is taken to be vanishingly small or else compensated). Then \(\varphi=\mu\gamma\). If \(X\) is sufficiently small in comparison with the reactance, which is easily attainable in practice, then the current in the induction circuit is determined by the expression

\[ -X\dot{\varphi}=-\gamma X\dot{\mu}. \tag{34} \]

Then the torque produced by the coil \(K\) will be equal to

\[ c=-\Gamma\gamma m_0 X\dot{\mu}. \tag{35} \]

If both torques \(c\) and \(g\) (equation 28) are equal in magnitude and opposite in phase, then the amplitude of the oscillations will become zero. This will occur for the value \(X=X_0\), determined from the equation

\[ -\rho=\Gamma\gamma m_0 X_0. \tag{36} \]

If a direct current \(i\) is passed through the coil \(K\), then the mirror turns through an angle \(\theta\), where \(\Gamma m_0 i=A\theta\), where \(A\)—

constant of torsion of the suspension. With measured \(\theta\), \(A\), and \(i\), one can eliminate \(\Gamma m_0\) from equation (35), after which one obtains

\[ -\rho=\gamma X_0 A\theta\cdot \frac{1}{i}. \tag{37} \]

In the experiments of Sucksmith and Bates, \(\Gamma m_0\) was eliminated precisely in this way; in the author’s experiments, \(\Gamma\) and \(m_0\) were carefully measured.

If the magnetizing coil is wound on the rotor, as shown in the figure, and compensation of mutual induction is absent, then instead of (35) one must use the equation

\[ -\rho=\Gamma\gamma m_0 X_0 \frac{1+\dfrac{\mu'}{\mu}} {1+\dfrac{\rho'\mu'}{\rho\mu}}, \tag{38} \]

where \(\mu'\) is the moment of the winding produced in air, and \(\rho'\) is the gyromagnetic ratio for an electron moving in a circular orbit, i.e. \(2\dfrac{m}{e}\).

In most of the author’s work the deflection method was used; its detailed development led to various modifications and additions, set forth in the following section.

§ 29. Theory of the complex deflection method

Let the first harmonic of the magnetic moment of the rotor be equal to \(\mu=\mu_0\sin\omega t\); then the first harmonic of the gyromagnetic moment \(-\rho\dot\mu\) will be equal to

\[ g=-\rho\omega\mu_0\cos\omega t=G\cos\omega t. \tag{39} \]

When \(\mu\) changes, an electromotive force \(\psi=-\omega\mu_0\gamma\cos\omega t\) is induced in the induction circuit, and a current \(\psi X=-\omega\mu_0\gamma X\cos\omega t\). This current, flowing around the coil \(K\), creates a moment

\[ c=-\omega\mu_0\gamma X\Gamma m_0\cos\omega t=C\cos\omega t, \tag{40} \]

acting on the rotor system. The total external moment \(\tau\) (frequency \(\nu=\dfrac{\omega}{2\pi}\)) acting on the system is equal to

\[ \tau=(G+C)\cos\omega t. \tag{41} \]

The frequency \(\nu\) of the first harmonic of the applied electromotive force can be made equal (or nearly equal) to the natural frequency of the oscillating system. Therefore the amplitude \(A\) of the oscillations can be represented by the expression

\[ A=\beta(G+C), \tag{42} \]

where \(\beta\) is a constant. The case is also possible

\[ A=\beta(G-C), \tag{43} \]

if (which is easy to realize) the coil \(K\) is connected into the circuit in such a way

so that \(c\) and \(q\) have opposite phases. In this case the amplitude becomes zero at the value \(X=X_0\), which ensures the equality \(G=C\), i.e. for \(\Gamma \gamma m_0 X_0=-\rho\).

A large part of the methods considered in §§ 27—30 have considerable advantages over other methods both with respect to finding and eliminating systematic errors caused by the influence of secondary moments, and with respect to the possibility of determining the sign of \(\rho\).

§ 31. Secondary rotating moments. Under practical conditions the gyromagnetic moment is never the only one present; it is accompanied by various secondary moments. A whole series of possible errors and methods for eliminating them was indicated in the work of Einstein and de Haas. Other errors have been taken into account in the works of other investigators and of the author.

A whole series of secondary moments can be eliminated if the oscillating body and the magnetizing coil are made strictly symmetrical (in the geometrical, mechanical, and magnetic respects) with respect to the vertical axis passing through the point of suspension. In all works much attention was paid to the fulfillment of these conditions. Since the complex resonance method has advantages over other methods with respect to detecting and eliminating systematic errors, and since most of the most accurate works were carried out by this method, it will be considered here first of all. Obviously, many conclusions will remain valid without any changes for other methods as well.

We shall take the variable magnetic moments of the rotor—the vertical and the horizontal—as respectively equal to

\[ \mu=\mu_0\sin\omega t \tag{49} \]

and

\[ \nu=\nu_0\sin(\omega t-\alpha). \tag{50} \]

We shall express the gyromagnetic moment by the equation

\[ g=G\cos\omega t. \tag{51} \]

We shall represent any secondary moment that is in the same or in the opposite phase with the gyromagnetic one by the equation

\[ p=P\cos\omega t. \tag{52} \]

If, however, it is in quadrature, it may be expressed as

\[ q=Q\sin\omega t. \tag{53} \]

The total moment will be written in the following form:

\[ t=g+p+q=\left[(G+P)^2+Q^2\right]^{\frac{1}{2}}\cos(\omega t-\delta) = T\cos(\omega t-\delta). \tag{54} \]

constant of torsion of the suspension. With measured \(\theta\), \(A\), and \(i\), one can eliminate \(\Gamma m_0\) from equation (35), after which one obtains

\[ -\rho=\gamma X_0 A\theta\cdot \frac{1}{i}. \tag{37} \]

In the experiments of Sucksmith and Bates, \(\Gamma m_0\) was eliminated precisely in this way; in the author’s experiments \(\Gamma\) and \(m_0\) were carefully measured.

If the magnetizing coil is wound on the rotor, as shown in the figure, and compensation of mutual induction is absent, then instead of (35) one must use the equation

\[ -\rho=\Gamma\gamma m_0 X_0 \frac{1+\dfrac{\mu'}{\mu}} {1+\dfrac{\rho'\mu'}{\rho\mu}}, \tag{38} \]

where \(\mu'\) is the moment of the winding, produced in air, and \(\rho'\) is the gyromagnetic ratio for an electron moving in a circular orbit, i.e. \(2\dfrac{m}{e}\).

In most of the author’s work the deflection method was used; its detailed development led to various modifications and additions, set forth in the following section.

§ 29. Theory of the complex deflection method

Let the first harmonic of the magnetic moment of the rotor be equal to \(\mu=\mu_0\sin\omega t\); then the first harmonic of the gyromagnetic moment \(-\rho\dot{\mu}\) will be equal to

\[ g=-\rho\omega\mu_0\cos\omega t=G\cos\omega t. \tag{39} \]

When \(\mu\) varies, an electromotive force \(\psi=-\omega\mu_0\gamma\cos\omega t\) is induced in the induction circuit, and the current is \(\psi X=-\omega\mu_0\gamma X\cos\omega t\). This current, flowing around the coil \(K\), creates the moment

\[ c=-\omega\mu_0\gamma X\Gamma m_0\cos\omega t=C\cos\omega t, \tag{40} \]

acting on the rotor system. The total external moment \(\tau\) (frequency \(\nu=\dfrac{\omega}{2\pi}\)) acting on the system is equal to

\[ \tau=(G+C)\cos\omega t. \tag{41} \]

The frequency \(\nu\) of the first harmonic of the applied electromotive force can be made equal (or almost equal) to the natural frequency of the oscillating system. Therefore the amplitude \(A\) of the oscillations can be represented by the expression

\[ A=\beta(G+C), \tag{42} \]

where \(\beta\) is a constant. The case is also possible

\[ A=\beta(G-C), \tag{43} \]

if (which is easy to accomplish) the coil \(K\) is connected into the circuit in such a way

so that \(c\) and \(g\) have opposite phases. In this case the amplitude becomes zero at the value \(X=X_0\), ensuring the fulfillment of the equality \(G=C\), i.e. when \(\Gamma \gamma m_0 X_0 = -\rho\).
The equation \(A=\beta(G-C)\) may be rewritten in the following form:

\[ A=\beta(G-\alpha X), \tag{44} \]

where, as shown in Fig. 13, the relations between \(A\) and \(C\) and \(A\) and \(X\) are linear.

The phases of the motion and of the moment change sign simultaneously at \(X=X_0\); if, however, we confine ourselves only to determining absolute values, then the relation between \(A\) and \(X\) is determined by two straight lines intersecting at \(X=X_0\) (Fig. 13 corresponds to the straight lines \(AF\) and \(FB\) in Fig. 14).

Fig. 13.                Fig. 14.

Depending on whether the external moment with amplitude \(Z\) is in phase with \(g\) or differs from it in phase by \(180^\circ\), the straight line \(G\) is replaced by the straight lines \(G+Z\) or \(G-Z\), intersecting the abscissa axis at the points \(X_0+\delta X_0\) or \(X_0-\delta X_0\). If such a moment exists and its phase can be changed to the opposite one without changing the amplitude, then the exact value \(X_0\) is determined as the mean of the two values \((X_0+\delta X_0)\) and \((X_0-\delta X_0)\).

If, however, there are moments in quadrature with \(g\), then the straight lines intersecting at \(X=X_0\) are replaced by the symmetric curve \(CED\) (Fig. 14), having a minimum at \(X=X_0\). Therefore the existence of such moments will not introduce an error into the determination of \(X_0\), provided only that the minimum of the amplitude \(EF\) is sufficiently small in comparison with the amplitude produced by \(g\); otherwise the curve near the minimum will become so flat that an exact determination of \(X_0\) will prove impossible.

§ 30. Various complex deflection methods

(A). Graphical method. The measured amplitudes \(A\) are plotted as a function of the conductance \(X\); the values \(X_0 \pm \delta X_0\) are determined from

position of the minimum (Fig. 14) or by the intersection of the abscissa axis with symmetrical straight lines satisfying the observations and extended to the abscissa axis (Fig. 13). In the presence of moments lying in quadrature, the experimental points near the intersection with the abscissa axis will lie outside the straight lines.

(B). Methods of large deflections. a) In many of the author’s works the values \(X_0 + \delta X_0\) and \(X_0 - \delta X_0\) were determined by carefully measuring the amplitudes \(A\) and \(A_2\), corresponding to \(X = 0\) and \(X = 2 (X_0 + \delta X_0) = X_2\); in addition, the amplitude \(A_0\) was measured approximately for \(X = X_0 + \delta X_0\).

The determination was made by the formula

\[ X_0 + \delta X_0 = X_2 \frac{(A^2 - A_0^2)^{\frac12}} {(A^2 - A_0^2)^{\frac12} + (A_2^2 - A_0^2)} \tag{45} \]

which is derived without any difficulty.

b) In another of the author’s works, the determination of \(X_0 + \delta X_0\) was carried out by finding half the value of \(X\) at which the precisely measured amplitude (far from the minimum) was equal to the precisely measured value of the amplitude at \(X = 0\). The first value was not observed directly, but was obtained by interpolation or extrapolation from values of \(X\) close to \(2 (X_0 + \delta X_0)\); the value \(X_0 - \delta X_0\) was determined in an analogous way.

c) In another method, less accurate and less frequently used, a comparison was made of the amplitude \(A_G\), arising only under the action of the gyromagnetic moment (the current through the coil \(K\) is absent, \(C = 0\)), with the amplitude \(A_C\), obtained in the presence only of the moment \(c = C \cos(\omega t + \delta)\), produced by the coil \(K\), while the current in the magnetizing coil is absent (\(G = 0\)). If the amplitudes measured in the two cases are respectively \(A\) and \(A'\), then we obtain:

\[ A_G = (A^2 - A_0^2)^{\frac12} \quad \text{and} \quad A_C = A' \tag{46} \]

and

\[ \frac{A_G}{A_C} = \frac{G}{C} = \left|\frac{\rho \omega \mu}{\Gamma m_0 C}\right|, \tag{47} \]

whence

\[ |\rho| = \frac{\Gamma m_0 C}{\omega \mu_0}\,\frac{A_G}{A_C}. \tag{48} \]

This method is applicable only in the case when it is known that the value \(A_0\) may be neglected, or else when \(A_0\) is determined by some other method.

A large part of the methods considered in §§ 27—30 have significant advantages over other methods both with regard to finding and eliminating systematic errors caused by the influence of parasitic moments, and with regard to the possibility of determining the sign of $\rho$.

§ 31. Parasitic rotating moments. Under practical conditions the gyromagnetic moment is never the only existing one; it is accompanied by various parasitic moments. A whole series of possible errors and methods for eliminating them are indicated in the work of Einstein and de Haas. Other errors have been taken into account in the works of other investigators and of the author.

A whole series of parasitic moments can be eliminated if the oscillating body and the magnetizing coil are made strictly symmetrical (in the geometrical, mechanical, and magnetic respects) with respect to the vertical axis passing through the point of suspension. In all works great attention has been paid to the fulfillment of these conditions. Since the complex resonance method has advantages over other methods with regard to detecting and eliminating systematic errors, and since most of the most accurate work was performed by this method, it will be considered here first. Obviously, many conclusions will, without any changes, also be valid for other methods.

Let us take the alternating magnetic moments of the rotor—the vertical and the horizontal—to be respectively equal to

\[ \mu=\mu_0\sin\omega t \tag{49} \]

and

\[ \nu=\nu_0\sin(\omega t-\alpha). \tag{50} \]

We shall express the gyromagnetic moment by the equation

\[ g=G\cos\omega t. \tag{51} \]

We represent some parasitic moment, which is in the same or the opposite phase with the gyromagnetic one, by the equation

\[ p=P\cos\omega t. \tag{52} \]

If, however, it is in quadrature, it may be expressed as

\[ q=Q\sin\omega t. \tag{53} \]

The total moment is written in the following form:

\[ t=g+p+q=\left[(G+P)^2+Q^2\right]^{\frac{1}{2}}\cos(\omega t-\delta) = T\cos(\omega t-\delta), \tag{54} \]

where

\[ \tg\delta=\frac{Q}{G+P}. \tag{55} \]

Equation (54) shows that the total moment, and with it the amplitude, increase in the presence of a moment that is in quadrature with the gyromagnetic one. In § 43 a special experimental method is described in which this moment has no influence on the motion of the system.

The most essential secondary moments may be divided into four classes, described in the following paragraphs.

§ 32. Moments caused by the terrestrial magnetic field

This field can be almost completely compensated by means of current-carrying coils or magnets placed near the apparatus. But there always arises the doubt whether the compensation is sufficiently perfect that the moments produced by the remaining part of the terrestrial field may be disregarded. However, their influence can be taken into account or made sufficiently small.

Let \(\Delta H\) characterize the intensity of the horizontal component of the residual terrestrial field in the space occupied by the rotor; let \(\Delta X\) and \(\Delta Y\) be its northern and eastern components; further, let \(\Delta Z\) be the vertical component of the terrestrial field. In this case the following combinations arise:

a) The axial moment acting on the rotor owing to the presence of the field \(\Delta H\) is equal to

\[ e_h = h \cdot \Delta H \cdot \nu = E_h \sin(\omega t-\alpha). \tag{56} \]

If the suspension system is turned through \(180^\circ\) about its axis, the magnitude of this moment remains the same, while its sign changes; therefore its influence can be taken into account (§ 29).

b) The field \(\Delta H\) produces a moment \(l = L \sin \omega t\) about the horizontal axis. If the system is not entirely symmetric, then this moment, together with the moment due to the displacement of the suspension and of the rotor relative to their supports, will lead to the appearance of an axial moment

\[ b = B \sin(\omega t-X), \tag{57} \]

which, like the moment \(e_h\), changes sign when the azimuth is changed by \(180^\circ\).

c) The residual vertical component, acting on the magnetic moment \(\nu\), will produce an effect similar to the effect considered in the preceding item, but the sign of this effect will not change when the azimuth is changed. Therefore direct elimination of this effect is possible only under the condition that \(\Delta Z\) or \(\nu\) is made sufficiently small. In the ballistic method the action of this moment is either added to the action of the gyromagnetic moment, or subtracted from it. The error introduced by it decreases when the mean result is determined from a large number of observations in which the rotor is repeatedly set in positions differing from one another only for random reasons,

In the resonance method this procedure makes it possible to eliminate the influence under consideration only in the case where the influence of the moment that is in quadrature with the gyromagnetic one has been eliminated or is sufficiently small.

d) If the magnetizing coil is wound on the rotor, then a moment arises

\[ i = I \sin(\omega t + \gamma), \tag{58} \]

where, for a rectangular form of the current curve, \(\gamma\) is very small; this moment retains its magnitude, but changes sign when the system is turned through \(180^\circ\).

§ 33. Moments caused by the influence of the coils on the moving system

a) If the magnetizing coil is fixed relative to the earth, and the rotor has a constant horizontal moment \(\xi\), then, generally speaking, an axial moment \(d\) may arise, acting on the rotor and caused by the horizontal component of the coil field, since the direction of its field will not be parallel to the direction of \(\xi\). This moment almost coincides in phase with \(\mu\), so that one may write

\[ d = D \sin(\omega t + \delta), \tag{59} \]

where \(\delta\) is a small quantity. This moment changes sign when the azimuth is changed by \(180^\circ\).

b) In the same way there arises an axial moment caused by the influence of the horizontal component of the alternating field and the alternating horizontal moment \(\nu\). If the half-periods of the magnetizing current coincide, then the fundamental frequency of the moment is twice the frequency of the current, and therefore its influence may be neglected. If, however, the half-periods do not coincide, then a residual moment arises whose frequency coincides with the frequency of the current. This moment is in quadrature with the moment \(g\), as was shown by de Haas.

c) In an analogous way there arises a moment about the horizontal axis of the rotor, caused by the action of the vertical component of the alternating field on the constant horizontal moment \(\xi\). It produces a relative displacement of the rotor and suspension; as a result of this, owing to incomplete symmetry with respect to the vertical, a moment about the vertical is produced, which may have any phase relative to \(g\). It has a frequency equal to the frequency of the current, and changes neither its magnitude nor its sign when the azimuth of the system is changed by \(180^\circ\).

d) The coil \(K\) creates a magnetic field penetrating into the space occupied by the rotor, where the field lines have a horizontal direction. In this case an axial moment may arise if the magnetization of the rotor is not quite symmetric with respect to the suspension axis.

If the magnetization is purely alternating (equal half-periods), the frequency of this moment is twice the frequency of the current, and it may be disregarded in the resonance method. However, if there exists a residual moment, normal to the axis of the coil, produced by the uncompensated part of the earth’s field or by some other cause, then an axial moment arises, having the frequency of the current and coinciding in phase (or differing by \(180^\circ\)) with the gyroscopic moment. In the author’s work described here this influence was vanishingly small. Its sign changes upon reversing the azimuth.

e) If the induction solenoid is not strictly vertical, but makes a small angle \(\alpha\) with the vertical, then its field has a horizontal component equal to \(h=\alpha\gamma\) per unit current. A moment then arises, acting on the rotor, which has magnetic moments \(\upsilon\) and \(\xi\). The influence of one of these moments disappears owing to the double frequency, while the other may be eliminated by reversing the azimuth of the rotor. A moment similar to the latter is also produced by the action of the induction current on the small magnet with moment \(m_0\).

§ 34. Moment due to the stray fields of the magnetizing and induction circuits

If such leakage exists, then a moment acts on the oscillating system,

\[ \lambda=\Lambda \sin(\omega t+\gamma). \tag{60} \]

The coefficient \(\lambda\) can be made very small by using good (magnetic) insulation, and \(\gamma\) can be reduced by using a rectangular current curve. Therefore this moment is small and is in quadrature with \(g\).

§ 35. Moment due to the existence of mutual induction between the magnetizing coil and the induction solenoid

This moment can be made vanishingly small if a mutual-induction compensator is used; otherwise its influence can be calculated, and a corresponding correction can be introduced into the results. In the author’s work, where the magnetizing coil was fixed, compensation was employed. If, however, the magnetizing coil is mounted on the rotor, then the small effect produced by it can be calculated and the corresponding correction introduced.

§ 36. Moment due to the inertia of the electrons in the rotor and in the rotor coil \(^{21}\)

If the strength of the current changes in the coil wound on the rotor, and with it also the angular momentum of the free electrons, then there arises

equal and opposite change in the moment of the rotor. Thus a moment is produced, due to the inertia of the electrons. It can be shown that it is in phase with the gyromagnetic moment \(g\), and that its amplitude \(T\) is related to the amplitude \(G\) of the gyromagnetic moment by the relation

\[ \frac{T}{G}=\frac{2m}{e\mu'}\frac{1}{\rho\mu}. \tag{61} \]

Its influence can be calculated exactly (§ 28). As for the moment due to the inertia of the electrons in the rotor, where induction currents are present, it is vanishingly small.

§ 37. Moment due to magnetostriction

Owing to magnetostriction, vertical motions are produced which, in the presence of asymmetry, or owing to a change in the torsion of the suspension under tension, may partly pass into vibration about the axis. Since the changes in the length of the rotor due to magnetostriction do not depend on the direction of the field, no moment can arise whose frequency coincides with the frequency of the first harmonic of magnetization, provided only that the two half-periods of magnetization are identical. However, if they are not identical (either owing to a residual vertical magnetic moment, or because of the asymmetry of the half-periods of the magnetizing current), then a component arises whose frequency coincides with the frequency of the first harmonic of magnetization. It may have any phase relative to the gyromagnetic moment.

If the half-periods of the current producing positive and negative magnetization of the rotor are interchanged, or if the ends of the magnetizing coil are switched without changing the direction of the current in the rest of the circuit, then this moment changes sign but retains its magnitude.

In the compound resonance method, where errors from the moment that is in quadrature with the gyromagnetic moment are eliminated, this effect can be eliminated by carrying out a series of measurements in which these switchings are repeated periodically. In the ballistic method this moment is either added to the gyromagnetic one or subtracted from it; the error can be reduced by determining the mean value from many measurements in which the rotor is repeatedly placed under the same conditions. In the case of the simple resonance method this procedure makes it possible to eliminate the error only in the case when the moment that is in quadrature has been destroyed or is sufficiently small.

§ 38. Earlier investigations of rotation under magnetization

The first experiments that gave some results were carried out by Einstein and de Haas \(^{22}\) in 1915. They worked with iron

resonance methods (1 and 2, § 26). For the value of $\rho$ they obtained a quantity close to $2\frac{m}{e}$; the accuracy of the observations, according to the authors’ estimate, was about 10%. The authors believed that they had also succeeded in determining the sign, which proved to be negative. However, Lorentz[^23] soon pointed out that the authors’ experiments were insufficient for determining the sign. In 1916 Einstein and de Haas,[^24] working independently of one another, carried out a resonance experiment (purely qualitative) at very low frequencies (1–2 Hz); they found that $\rho$ is negative and in order of magnitude agrees with the value obtained by them in their first work.

Einstein and de Haas’s work was confined to comparatively few measurements with iron. The first detailed investigation of this effect in iron and nickel by the ballistic method was performed in 1918 in the Richardson laboratory (Princeton) by Stewart,[^25] who found for iron $\rho = 1.02 \frac{m}{e}$ and for nickel

\[ \rho = 0.94 \frac{m}{e} \]

with a mean error of 15%; this value agrees (within the experimental errors) with the values obtained by the author of the present article in 1914 in work on the magnetization of iron during rotation, and constitutes half the value found by Einstein and de Haas. This investigation was one of the few in which the vertical component of the earth’s magnetic field was completely compensated and in which much effort was devoted to eliminating the effect of magnetostriction.

In 1919 Beck,[^26] working by resonance methods (1 and 2), found $\rho = 1.06 \frac{m}{e}$ for iron and $\rho = 1.14 \frac{m}{e}$ for nickel. But the errors of measurement were so large that he considered the most probable values of $\rho$ to be those coinciding with $\frac{m}{e}$. In the same year Ervidson,[^27] working by the resonance method (2) and eliminating all components of the earth’s field, found for iron $\rho = 0.94 \frac{m}{e}$.

§ 39. Later experiments with the rotation of ferromagnetic substances during magnetization

(The experiments of Chattock and Bates)

Chattock and Bates[^28] in 1923 published the results of an investigation carried out by the ballistic method and surpassing, in the thoroughness of its execution and the reliability of its results, the works mentioned above. They devoted great attention to eliminating possible systematic errors, but the vertical component of the earth’s field was not compensated by them. For iron and nickel they obtained mean values $\rho \frac{e}{m}$ equal, respectively, to 1.005 and 1.01. The value of $\rho$, calculated by formula (31) from the known

was found to be a linear function of \((1+\lambda)\), as shown in Fig. 15. The upper curve \(B\) represents the results of observations at a large value of the system’s moment of inertia (proportional numbers 7–15, placed near the curve). The lower curve \(A\) corresponds to the smaller moment of inertia (from 1 to 3.6 in the same units). The authors considered that the exact value of \(\rho\) would be obtained by extrapolating these curves to \(\lambda=0\). The observations were very numerous; each point of the graph characterizes the mean value for 105 readings for each value of \((1+\lambda)\). The mean scatter of the points for group \(B\) is \(2\%\), and for group \(A\), \(1.5\%\). The discrepancies between individual observations were more significant. It was found, as in other investigations, that the quantity \(\rho\) does not depend on the intensity of the magnetizing field.

§ 40. The Experiments of Sucksmith and Bates and the Experiments of Sucksmith

In the same year there appeared another detailed investigation of iron, nickel, and Heusler alloy, in which Sucksmith and Bates \(^{29}\), working by Chetcock’s null method, obtained for these substances the following values: \(\rho = 1.006;\ 1.002;\ 1.002\,\dfrac{m}{e}\). In 1925 Sucksmith \(^{30}\), investigating cobalt and magnetite by the same method, obtained the values \(\rho = 1.03\) and \(0.99\,\dfrac{m}{e}\). The authors believed that in the first of these investigations the accuracy reached \(1\%\) and that, consequently, \(\rho\,\dfrac{e}{m}\) is equal to unity. The most important results of these works are given in Table 3.

Fig. 15.

Table 3

The gyromagnetic ratio according to the works of Sucksmith
and of Sucksmith and Bates

\[ \left(\frac{e}{m}=1.77\cdot 10^7\ \mathrm{CGSM}\right) \]

Substance	Frequency (s\(^{-1}\))	Field intensity (oersted)	Series No.	Values \(\rho\cdot \dfrac{e}{m}\)	Amplitude of angular oscillation (mm)
Iron	31—78	90	16	0.974—1.028 1.006 ± 0.012	10—20
Nickel	38—50	45—176	6	0.988—1.010 1.002 ± 0.004	5—10
Heusler alloy	21—40	176	14	0.988—1.014 1.002 ± 0.006	5—8
Cobalt	33—46	225	9	0.916—1.160 1.030 ± 0.064	1
Magnetite	27—59	225(?)	10	0.894—1.146 0.990 ± 0.048	0.7

Note. Sucksmith recently obtained (Nature, 134, 936, 1934) for three nickel alloys (about 56% nickel) near the Curie point

\[ \rho\cdot \frac{e}{m}=1.05\pm 10\%. \]

In the last two cases rods of pressed powder were used, 15.2 cm long; their diameters varied from 1.65 to 3.4 mm.

As in the work of Chattock and Bates, in these works great attention was paid to the elimination of systematic errors; but the vertical component of the earth’s field was not compensated, and the effect of magnetostriction was not taken into account. In the author’s work it was shown that in some cases these effects may play an essential role. The data published in these works are insufficient for the reader to form a definite impression of the magnitude of the errors introduced by these factors.

§ 41. Barnett’s experiments on rotation during magnetization

Taking into account the discrepancy between the works described above and the results obtained by Barnett in studying magnetization during rotation, the author continued the investigation of the Einstein—de Haas effect, begun several years earlier but temporarily interrupted because of the abundance of urgent current work.

In this carefully developed investigation a complex resonance method was used. Great attention was paid to the inve—

... to the investigation and elimination of systematic errors. The results of the determination of \(\rho\) for the most reliable series of observations are given in Table 4.

Table 4

Gyromagnetic ratio from Barnett’s work on rotation during magnetization

Substance	Frequency (hertz)	External magnetic field (oersted)	Series No.	\(\rho \cdot \dfrac{e}{m}\)
Armco iron	10	15—30	25	\(1,031 \pm 0,003\)
Electrolytic iron	9	20	7	\(1,031 \pm 0,007\)
Electrolytic iron	9	20	2	\(1,028 \pm 0,002\)
Nickel	5,4—12,8	40	60	\(1,05 + \pm < 0,01\)
Permalloy (80% Fe, 20% Ni)	10	20	17	\(1,043 \pm 0,003\)
Permalloy (80% Fe, 20% Ni)	19	35	10	\(1,041 \pm 0,005\)
Iron–nickel (75% Fe, 24,5% Ni)	5	40	8	\(1,015 \pm 0,012\)
Cobalt (99,9%)	10	40	5	\(1,096 \pm 0,002\)
Cobalt (99,9%)	10	40	5	\(1,081 \pm 0,016\)
Copper–cobalt (92,4% Co)	5,5	40	12	\(1,068 \pm 0,006\)
Cobalt–nickel (54% Co, 45% Ni)	9	20	4	\(1,087 \pm 0,038\)
Cobalt–iron (34% Co)	3	20—40	5	\(1,009 \pm 0,007\)

Each series, in addition to the large number of readings necessary for determining the values of the current that compensated the earth’s magnetic field and for other auxiliary measurements, contained 48, 72, or 96 determinations of the amplitude, which were made according to a developed schedule; all measures were taken for the fullest possible elimination of all possible systematic errors considered above.

The mean value of \(\rho \dfrac{m}{e}\) for all the substances investigated was \(1,047\), which is in excellent agreement with the mean result of the investigation of the Barnett effect (§§ 22, 23).

Earlier series (41 in number) of investigations of permalloy and iron, carried out by the same method but without such strict observance of the equality of the observation time, gave almost the same mean result \(\left(\rho \frac{e}{m} = 1.050 \pm 0.013\right.\) for permalloy and \(1.034 \pm\)

\(\pm 0.008\) for iron). More than 150 series obtained before careful measures had been taken to exclude the influence of circular asymmetry gave almost the same mean result.

Since the error in the determination of \(\rho\) depends on the accuracy of the measurement of the constants, all of them were repeatedly measured with the greatest possible care; thanks to this the error in the determination of \(\rho\) did not exceed 0.5%.

In almost all the measurements whose results are given in the table, and in most other measurements, the magnetizing solenoid was wound rigidly on the rotor, so that the total moment could be measured very accurately. Experiments with electrolytic iron, in which the magnetizing coil was fixed immovably relative to the earth, give results agreeing with the others. In addition, 31 further series of observations were made at four different values of the current, the coil being alternately movable or fixed. The results showed a small systematic discrepancy, the cause of which has not yet been clarified; the gyromagnetic ratio for the movable coil systematically exceeded the ratio for the fixed coil by about 1%. Later, but shorter, series gave a similar unexplained difference, but of the opposite sign. A large number of other observations, including many measurements with iron and permalloy made in the initial stage of the investigation with a fixed coil, are in close agreement with the other observations. But these observations are obviously less free from errors, because they were made before measures had been taken to eliminate from each series the influence of circular asymmetry.

Almost all the measurements were made after midnight, when all possible disturbances are minimal. The apparatus was installed on a spring support and provided with damping devices.

The construction of most of the rotors used is shown in Fig. 16. The magnetic rod was 20 or 27.5 cm long; its diameter ranged from \(1/16\) to \(1/8\) inch. Some rotors had no winding. A third, more complicated, construction was used in some cases in order to reduce the magnetostrictive effect (Fig. 17). The magnetic rod was mounted in a coaxial brass tube of somewhat greater length. Three brass rings, fitted with their inner surface to the rod and with their outer surface to the tube, were soldered to the rod at the ends and in the middle. The middle ring was also soldered to the tube. In this way the middle of the rod was rigidly connected with the tube, while the ends could

slide along it. The magnetizing coil was wound on a brass tube. This construction was developed on the basis of the consideration that, when magnetostrictive oscillations arise, the center of gravity of the rod will remain stationary, and the oscillations will not affect the brass tube and the suspension.

Fig. 16.

• A — suspension
• B — upper end (brass)
• C — Bakelite ring
• D — magnetic material
• E and F — magnetizing coil
• G — magnetic material
• H — lower end (brass)
• I
• J

Fig. 17.

• 45.6 cm length; wire
• \(\varphi = \frac{1}{32}''\)
• washer
• \(\varphi = \frac{1}{4}''\)
• \(\varphi = \frac{9}{32}''\)
• ring soldered only to the rotor
• \(\varphi = \frac{9}{32}''\), cobalt rotor
• \(\varphi = \frac{5}{32}''\), brass tube
• rotor soldered to the tube
• Note: winding of copper wire (4 layers), wound on the tube
• ring soldered only to the rotor
• \(\varphi = \frac{1}{4}''\)
• \(\varphi = \frac{1}{8}''\)
• 48″ (121.5 cm)
• 1″

§ 42. Sucksmith’s Experiments with Paramagnetic Substances

Sucksmith \(^{32}\) used the resonance method (1) to investigate mixtures of certain elements—rare earths (\(\mathrm{Cd_2O_3}\), \(\mathrm{Nd_2O_3}\), \(\mathrm{Eu_2O_3}\), \(\mathrm{Dy_2O_3}\)) and salts of the iron group (\(\mathrm{FeSO_4}\), \(\mathrm{CoCl_2}\), \(\mathrm{CoSO_4}\), \(\mathrm{CrCl_2}\), \(\mathrm{MnCO_3}\), \(\mathrm{MnSO_4}\)). The magnetic moments that arose in the specimens under investigation, even in the strongest fields (up to 1200 oersteds), were so small that it was necessary to reduce the frictional moment by using very low frequencies (less than 0.3 Hz, smoothed curve), making the suspension from fused quartz and placing the oscillating body in an evacuated space. Even under these conditions the double amplitude of the oscillations was very small, ranging from 1 to 4 mm. In addition to the precautions taken by Sucksmith and Bates when working with ferromagnetic bodies, measures were taken here to

GYROMAGNETIC EFFECTS AND EFFECTS OF ELECTRON INERTIA

eliminating the influence of ferromagnetic impurities, as well as of electric charges.

From Sucksmith’s experiments values of \(\rho\) were obtained, as well as values of the Landé splitting factor

\[ g = 2\frac{m}{e}\cdot\frac{1}{\rho}. \]

These values agree well with the values of \(g\) calculated on the basis of the most probable theoretical considerations concerning the state of the ions, which in some cases is insufficiently clear.

For other cases the results coincide with the data of Hund’s theory, as modified by Van Vleck \(^{33}\) for the rare earths, and with the data of Stoner’s theory \(^{34}\) for the iron group; the agreement (within the experimental error—5–10%) is quite satisfactory.

§ 43. Experiments of Ketterer and Scherrer with Pyrrhotite and Iron \(^{35}\)

In these experiments currents with smoothed curves and a long period (of the order of 5 sec.) were used, and sharp resonance was automatically produced and maintained (with the aid of a photoelement and a polarized relay). The photoelement was controlled by a very narrow beam of light reflected from a mirror connected with the oscillating rod, and falling on the photoelement only at those moments when the rod passed through its equilibrium position. By means of the relay, the photoelement changed the direction of the current at these moments. The advantage of such an arrangement consists not only in producing sharp resonance, but also in automatically eliminating the influence of torques in quadrature with the gyromagnetic torque. The authors give almost no details of the experiment. The specimens were suspended on a quartz thread and oscillated in vacuum.

Thirty measurements were made with four specimens of pyrrhotite. They were prepared by packing the powdered substance into narrow tubes placed in a magnetic field parallel to the direction of its intensity, so that the particles themselves were arranged along the direction of easiest magnetization. As Weiss showed, pyrrhotite is paramagnetic everywhere except in one plane. The following values of \(g\) were obtained: 0.62, 0.63, 0.63, 0.64.

This result may be interpreted as follows \(^{36}\): let us consider one or several independent \(d\)-electrons. If saturation occurs, then in the state corresponding to the minimum energy the orbital moment will be antiparallel to the spin. But

\[ m_s=-\frac{1}{2} \]

and \(m_l=2\). Therefore one obtains

\[ g=\frac{2}{3}. \]

Incomplete orientation may reduce the value of \(g\), which according to Ketterer is equal to 0.63. However, the published data do not make it possible to decide whether this discrepancy is real or is due to experimental errors.

According to Ketterer and Scherrer’s data for iron,

\[ g=2.01, \]

or

\[ \rho=0.995\frac{m}{e}. \]

For iron powder Ketterer \(^{37}\) obtained

\[ \rho=1.01\frac{m}{e}. \]

S. Barnett

Figure 18 shows (as a function of time) the magnetic moment produced by a rectangular current curve, and its harmonic—a sinusoidal curve, denoted by \(\mu\). The gyroscopic moment is, more properly, the moment summing with a certain (coinciding or opposite in phase) term and the curve \((g+p)\), differing from \(\mu\) by \(1/4\) of a period. In agreement with the experimental data, let us assume that the damping is proportional to the angular velocity of the rotor \(\dot{\theta}\). Then we obtain

Fig. 18.

\[ \dot{\theta}=a(g+p)=\Theta\cos\omega t \tag{62} \]

and it coincides in phase with \((g+p)\). In the experiments of Ketterle and Scherrer this relation between \(\dot{\theta}\) and \((g+p)\) was evidently maintained by the method described above. Since the mean value of any moment in quadrature is \(q=Q\sin\omega t\), it follows that

\[ \overline{q\dot{\theta}}=\overline{Q\Theta\sin\omega t\cos\omega t} =\frac{Q\Theta}{2}\,\overline{\sin 2\omega t}=0. \tag{63} \]

Thus a moment in quadrature will have no effect on the motion.

§ 44. Ray-Chaudhuri’s Experiments[^38] with Iron Oxides

These experiments were carried out by the resonance method (1), modified by Sucksmith for work with paramagnetic substances (§ 42). The substance in powder form was pressed into glass tubes from 1.3 to 2.3 cm in diameter and 6 cm long. The cylinders oscillated in vacuum in a magnetic field of strength 230 oersteds. Thin cylinders gave a spot displacement of 3 cm at a distance to the scale of 1.3 m. The probable error was estimated at approximately 2%. The details of the experiment are scarcely described, and therefore the reader is deprived of the possibility of estimating the error independently. For \(\rho\,\frac{e}{m}\) the following values were obtained: \(\mathrm{Fe_3O_4}\) (precipitated) 1.008; \(\mathrm{Fe_2O_3}\) (ferromagnetic) 1.016; \(\mathrm{NiO}\ \mathrm{Fe_2O_3}\) 1.022.

E. Gyroscopic Magnetization in a Rotating Field

§ 45. Gyroscopic Magnetization in a Rotating Field (see § 3)

The gyromagnetic effects considered in the preceding paragraphs provide valuable data on the nature of magnetic elements, but

do not at all clarify the question of the magnetization process, which remains as unclear as before.

However, there are indications that in the initial stage of the process, i.e., in weak fields, magnetization proceeds by jumps corresponding to the rotation of individual magnetic elements, and not by a continuous change in their orientation, as required by the classical theory. On this basis Einstein predicted a null effect, supposedly confirmed by the experiments of the author and Fisher described in § 3. However, in 1925 the author showed that the value of the magnetization expected by Fisher, arising during rotations, should be multiplied by a small coefficient equal to \(3/2\) of the ratio of the transverse magnetization to the magnetization at saturation. The errors in Fisher’s experiment precisely covered this reduced possible effect.

It is possible that this reduction should be still more significant, since the opposing moments that arise may be much larger in the case when the magnetic elements rotate relative to the rest of the atomic structure than when they rotate together with it.

Fisher’s principal experiments were carried out with powdered iron and magnetite and with solid magnetite. The author’s principal experiments were carried out with pressed dust of iron and permalloy. In both works a magnetometric method was used, similar to the method employed by the author in his earlier works on magnetization during rotation.

In the author’s work an external magnetic field of 15 oersteds was used, rotating 15,000 times per second; in this case the average deflection of the magnetometer when the direction of rotation of the field was changed (with all corrections taken into account) was about 0.2 mm (in the wrong direction), while the average error was about 1 mm. The deflection required by the improved theory should have been 20 mm for permalloy and 7 mm for iron. If, however, one proceeds from Fisher’s rotation hypothesis (or from the Barnett effect), then under these conditions deflections of 5000 and 3500 mm should have been obtained. Similar results were obtained with permalloy dust at a frequency of 21,000 Hz. In this case all systematic errors associated with residual magnetization of the rod, heating, leakage between the anode circuit of the tube generator and the magnetizing coil (they were coupled through a transformer), and eddy currents were eliminated.

Part II. The Inertia Effect of Electrons

A. Introduction

§ 46. Maxwell’s Works and Those of His Followers

In Maxwell’s Treatise on Electricity and Magnetism, §§ 574, 575, and 577 describe three inertial effects which

must exist in conductors if an electric current is produced by the motion of only one kind of electric charge and if these charges possess inertia.

1. If the current is varied in a circular conductor or in a circular cylindrical coil that can freely move about its axis, then the charges will acquire some acceleration, and the coil itself (or the conductor) must acquire an acceleration in the opposite direction; moreover, the changes in the torques must be equal in magnitude but opposite in sign. Following Maxwell, Lodge sought this effect (1892) ^39; but it was found only by the author in 1930 (§§ 55–58). ^40

2. If a coil is traversed by a steady electric current and the charges possess a constant moment relative to the axis of the coil, then the coil must have the properties of a gyroscope. Maxwell sought this effect in 1861, but without success, since the experimental difficulties were too considerable (§§ 3 and 7). The author’s experiments, devoted to the study of the magnetization of iron during rotation (§ 3 and §§ 8–23), revealed this effect for individual Ampèrean eddy currents, each of which behaved as Maxwell’s coil would have had to behave if it had been possible to create the appropriate conditions.

3. If a coil acquires an acceleration relative to its axis, then the free charges will also be accelerated; moreover, when the speed of the coil increases they will lag somewhat behind, and when the coil slows down they will run ahead. Therefore the acceleration of the coil must be accompanied by the appearance of a current in it. This effect was discovered by Tolman and Stewart in 1918 and studied in four papers by Tolman and Stewart ^41, Tolman, Karrer, and Guernsey ^42, and Tolman and Mott-Smith ^43 (§§ 49–53). Before the appearance of the work of Tolman and Stewart, this effect had been investigated by the centrifugal method in the works of Lebedev (§ 47) and Nichols (§ 48).

B. The Centrifugal Experiments of Lebedev and Nichols

§ 47. Lebedev’s Experiments ^44

In these experiments, which were Lebedev’s last work and which represented only an introduction to a more extensive work interrupted by his death, toroidal rings of nonmagnetic materials (ebonite, brass, water, benzene) were set into rotation about their axis at a frequency of about 500 Hz. The rings were 2 cm thick; their inner and outer diameters were respectively 3 and 6 cm. Lebedev assumed that the positively charged part of the atom would not be displaced in the radial direction, whereas the electrons experience a centrifugal displacement proportional to a constant \(K\), depending on the nature of the material of the toroid, the radius, and the square of the frequency. Lebedev calculated the distribution of the convection current over the toroid. Further, he made a “current model” representing a toroid of the same dimensions as the one under investigation

this toroid consisted entirely of wire wound in such a way that, when an electric current was passed through the toroid, the latter would be distributed in the same way as the convection current in the toroid under investigation. The toroid under investigation and the current model were placed near an astatic magnetometer in such a way that their fields were directed identically, and a comparison was made of the deflections of the magnetometer arising under the action of the toroid or of the model.

Using his formula and Kelvin’s theorem of similitude, Lebedev showed that if in two symmetrical rotating bodies made of the same material two corresponding points have identical linear velocities, then the magnetic-field intensities created at two corresponding points of space must be equal to one another and proportional to the cube of the rotational speed of the bodies.

It follows from this that at the equator of a sphere 6 cm in diameter and making 500 rev/sec there should arise a magnetic field amounting to approximately one hundredth of the earth’s field (at the earth’s equator), if one assumes that the material of the sphere and of the terrestrial globe is the same and that the constant \(K\) is the same for both bodies, despite the fact that at the equator of the sphere the centripetal acceleration is almost 10 million times greater than on the earth. Lebedev supposed that, owing to this enormous difference in accelerations, the actual value of \(K\) for the sphere must decrease significantly.

Lebedev found that a current of 0.1 A in the current model creates a field equal to the earth’s field. The linear velocity at the equator of the toroid was 0.2 of the velocity at the earth’s equator. It follows from this that the magnetic-field intensity due to the rotation of the toroid corresponds to the intensity created by a current equal to \(0.1 \cdot (0.2)^3 = 0.001\) A, i.e., to approximately one hundredth of the intensity of the earth’s field. This current, flowing through the model, did indeed produce a deflection of the magnetometer by 10 divisions of the scale, but when one of the toroids was rotated it was not possible to obtain a noticeable deflection of the magnetometer.

§ 48. Nichols’ Experiments^45

If a metallic disk rotates about its axis with frequency \(\nu\) Hz, then free charges having magnitude \(e\) and mass \(m\) must be displaced toward the periphery of the disk until an equilibrium state is established in which, at a distance \(r\) from the center of the disk, an electric field \(E\) is created satisfying the condition

\[ E = \left(\frac{m}{e}\right) 4\pi^2 \nu^2 r . \tag{64} \]

Integrating this expression from the edge of the disk (radius \(R\)) to the center,

we obtain for the potential difference between the disk and the center the following expression:

\[ V=\left(\frac{m}{e}\right) 2\pi^2 R^2 \nu^2. \tag{65} \]

In Nichols’s experiments the disk was made of aluminum, \(R=10\) cm, \(\nu=100\) Hz and more. From equation (64) it follows that, if the free charges in the metal are electrons, then \(V=10^{-8}\) V and the field is directed from the center to the periphery; if, however, the free charges are protons, then \(V=2\cdot 10^{-5}\) V and the field is directed from the edge of the disk toward its center. By means of a galvanometer and brushes placed at the edge of the disk and near its center, Nichols attempted to measure this potential difference. In the case of protons a deflection of 2000 scale divisions should have been obtained; in the case of electrons, 1 division. In practice it was possible to observe irregular deflections of 500 divisions and more; distortions were introduced mainly by the thermal effect and by imperfections of the contacts. In any case, however, these experiments showed that the value of \(\frac{m}{e}\) for the true carriers of current in a metal is smaller than for protons.

C. The Ballistic Experiments of Tolman and Stewart \(^{41}\)

§ 49. Details of the Experiments

In these experiments a circular wire coil, tightly wound on a rigid nonmagnetic frame, was rapidly rotated about its axis, which was positioned vertically, and was suddenly braked by suitable brakes. The ends of the coil were connected to a galvanometer by means of a long cord, the authors having succeeded in overcoming the difficulties introduced by sliding contact. An auxiliary coil was included in the coil circuit; it had the same area and was placed parallel to the first, but was wound in the opposite direction. This arrangement made it possible to reduce to a minimum the influence of variations in the intensity of the earth’s field on the readings of the galvanometer.

The rotating coil was surrounded by another, constructed and oriented in such a way that, when current was passed through it, one could compensate the vertical component of the earth’s field in the space occupied by the rotating coil. This was necessary in order to eliminate the electromotive force which would have to arise in the coil upon its expansion during rapid rotation under the action of the centrifugal force. In addition, this electromotive force could be excluded from the observations when the direction of rotation of the coil was reversed, since it does not depend on the direction of rotation, whereas the effect observed upon reversal of the direction of rotation changes sign. For greater reliability the authors employed both of the described methods.

In making the observations, the coil was brought into the fastest possible rotation, and the zero reading of the galvanometer was recorded—

meter; then the brakes were applied and the coil stopped (in a fraction of a second), while the deflection of the galvanometer was noted.

§ 50. Theory of the Method and Results

The arrangement of the electrical part of the experiment is shown in Fig. 19. \(ABC\) is the rotating coil, \(D\) is the galvanometer, \(EFG\) is the compensating

Fig. 19.

coil. The arrow indicates the direction of traversal of the contour adopted as positive. Let \(l\) be the length, \(dl\) an element of length of the homogeneous conductor \(ABC\), \(s\) its cross-section, \(V\) its linear velocity (relative to the fixed coordinate system); \(c\) the current density, \(k\) the electrical conductivity of the conductor, \(i = kc\) the current in it, \(v\) the velocity of an electron in the conductor \(ABC\).

An electron of mass \(m\) and charge \(e\), located in some element \(dl\) of the conductor, is acted upon by the following two forces: the first, equal to \(Xe\), is due to the existence of the electric field \(X\); the other, equal to \(\dfrac{ce}{k}\), is due to the existence of friction. The equation of motion of the electron will be written as:

\[ Xe - \frac{ce}{k} = m\frac{dv}{dt}. \tag{66} \]

Dividing by \(e\) and integrating with respect to \(l\) (along the path \(ABC\)), we obtain, changing the signs of all terms:

\[ \frac{cl}{k} - \int Xdl = -\frac{ml}{e}\frac{dv}{dt}. \tag{67} \]

Let \(R_c\) and \(L_c\) denote the resistance and inductance of the coil \(ABC\); \(R_g\) and \(L_g\), the resistance and inductance of the remaining part of the circuit. Finally, let \(U\) be the potential difference on the section \(ABC\), equal to the potential difference on the section \(AFC\). We may write:

\[ -\int Xdl = -\left(U - L_c\frac{di}{dt}\right) \]

\[ = R_g i + L_g\frac{di}{dt} + L_c\frac{di}{dt}. \tag{68} \]

The first term in equation (67) is, obviously, equal to \(R_c i\). Therefore the equation may be given the following form:

\[ (R_c+R_g)i+(L_c+L_g)\frac{di}{dt}=-\frac{ml}{e}\frac{dv}{dt}. \tag{69} \]

If now the conductor is stopped and the same \(i\) and \(\dfrac{di}{dt}\) are produced in it by means of a suitable generator having an electromotive force equal to \(E\), then instead of equation (69) the following expression is obtained:

\[ (R_c+R_g)i+(L_c+L_g)\frac{di}{dt}=E. \tag{70} \]

Consequently, the acceleration of the electrons in the moving conductor creates an internal electromotive force

\[ E=-\frac{ml}{e}\frac{dv}{dt}. \tag{71} \]

The value of the integral of equation (69), taken within the limits from the instant when the coil and the electrons contained in it have the constant velocity \(v=V=V_0\), to the instant when the coil stops and the current in it ceases \((v=V=0)\), is evidently equal to

\[ RQ=\frac{ml}{e}V_0, \tag{72} \]

where \(R=R_c+R_g\) is the total resistance of the circuit, and \(Q=\int i\,dt\) is the total quantity of electricity that has passed through the galvanometer. For the ratio \(\dfrac{m}{e}\) we obtain:

\[ \frac{m}{e}=\frac{RQ}{lV_0}. \tag{73} \]

624 experiments were carried out with various coils, with three different kinds of wire (copper, aluminum, and silver) of two different sizes, and with two different methods of fastening the coil. The resistance of the circuit varied from 27 to 63 \(\Omega\), the length of the coil wire from 285 to 529 m, and the linear velocity from 19.8 to 56.4 m/sec. For each direction of rotation a whole series of measurements was made. The direction of the current was in all cases such that the current carriers had to be regarded as negative charges; for each substance, calculation by formula (72) gave results in good agreement. The values of \(\dfrac{m}{e}\), calculated in fractions of this ratio for a free electron, proved to be 1.11, 1.16, and 1.20 for copper, aluminum, and silver respectively. Since it was observed that an increase in the value ...

tion of \(m/e\), in comparison with its standard value, is obtained the smaller, the more rigidly the winding of the coil is made; therefore it was suggested that insufficient rigidity of the winding causes a systematic error increasing the measured value of \(m/e\). Below, in § 53, experiments are described which apparently confirm this suggestion.

D. The Experiments of Tolman, Karrer, and Guernsey and Tolman and Mott-Smith

§ 51. General plan of the experiments and theory

In each of the investigations described, of which the second was an extension of the first, two principal operations were carried out: 1) a round hollow copper cylinder, whose axis was situated practically parallel to the direction of the earth’s field, was set into harmonic oscillations along this axis, the amplitude of the oscillations being kept constant and equal to \(A\); the frequency of the oscillations was \(\nu=\frac{\omega}{2\pi}\) Hz. Owing to the inertia of the electrons, an alternating current arose in the cylinder. The cylinder was surrounded by a stationary coil coaxial with it, containing a large number of turns of fine wire. By means of the method described below, the electromotive force \(E_a\), induced in this coil by the current due to the inertia of the electrons, was measured; 2) a similar cylinder, likewise placed inside an analogous coaxial coil, was arranged so that its axis was perpendicular to the direction of the earth’s field and was set into oscillation along the axis, perpendicular to the field and to the axis of the cylinder, with the same frequency \(\nu\) and with amplitude \(B\). In the same way as in item 1, the electromotive force \(E_b\), induced in the coil by currents arising in the cylinder owing to electromagnetic induction, was measured.

It is evident that the distribution of the current density must in both cases be exactly the same and must coincide with the distribution of the electric field, since the lines of current and the lines of the electric field must be circles coaxial with the cylinder and the coil.

Let us mentally divide the cylinder into a series of coaxial tubes, each of which has an infinitely small thickness \(dr\), and consider one of these tubes, having a mean radius equal to \(r\). In the case described in item 2, the oscillations of the cylinder will produce in each tube of radius \(r\) an electromotive force \(E=-\frac{d\varphi}{dt}\), where \(\frac{d\varphi}{dt}\) is the quantity determining the increment of the magnetic flux \(\varphi\) through the tube of radius \(r\), caused by its motion in the earth’s field.

If by the letter \(i\) we denote the current (circular, cylindrical) flowing in the tube, and if \(R\) and \(L\) are its resistance and inductance, then the force acting on an electron of mass \(m\) and charge \(e\) will be equal to

\[ \left(E-L\frac{di}{dt}-Ri\right)\frac{e}{2\pi r}=m\frac{dv}{dt}. \tag{74} \]

If \(n\) is the number of free electrons in a unit volume and \(a\) is the cross-sectional area of the conductor, then

\[ i=neav. \tag{75} \]

Therefore the preceding equation may be rewritten in the following form:

\[ E=Ri+\left(L+\frac{2\pi rm}{ne^{2}a}\right)\frac{di}{dt}. \tag{76} \]

In the experiment described in item 1, when the oscillations of the cylinder occur exactly along its axis, the electromotive force \(E=-\dfrac{d\varphi}{dt}\) becomes zero. Therefore, for the same current \(i\) and for the same value of \(\dfrac{di}{dt}\), instead of equation (74), in this case there must be

\[ \left(-L\frac{di}{dt}-Ri\right)\frac{e}{2\pi r}=m\frac{dv'}{dt}, \tag{77} \]

where \(v'\) is the velocity of the electron. But, since in the present case the tube has a tangential velocity \(V\), we obtain

\[ i=nea(v'-V). \tag{78} \]

Eliminating \(v'\), we obtain from (74) the following expression:

\[ -\frac{2\pi rm}{e}\frac{dV}{dt} = Ri\left(L+\frac{2\pi rm}{ne^{2}a}\right)\frac{di}{dt}. \tag{79} \]

Comparing this equation with equation (74), we see that the acceleration of the conductor causes the appearance of an electromotive force

\[ E_{e}=-\frac{2\pi rm}{e}\frac{dV}{dt}, \tag{80} \]

proportional to the acceleration and in phase with it if \(e\) is negative, or opposite in phase if \(e\) is positive.

Let us now consider processes 1 and 2 as applied to our tube. In the first case, when the angular displacement \(\theta=A\sin\omega t\), we obtain

\[ E_{e}=-\frac{2\pi rm}{e}\frac{dV}{dt} = -\frac{2\pi r^{2}\omega^{2}mA}{e}\sin\omega t, \tag{81} \]

In the second case, when the angular displacement is \(\psi = B\sin(\omega t+\delta)\) and \(H\) is the full intensity of the earth’s magnetic field, we find

\[ \varphi=\pi r^{2}HB\sin(\omega t+\delta) \tag{82} \]

and

\[ E=-\frac{d\varphi}{dt}=-\pi r^{2}HB\omega\cos(\omega t+\delta). \tag{83} \]

Consequently, at some instant of time \(t\), for a tube of radius \(\rho\) one obtains

\[ \frac{E_e}{E}=\frac{2A}{B}\frac{m}{e}\frac{\omega}{H}\frac{\sin\omega t}{\sin(\omega t+\delta-90^\circ)}; \tag{84} \]

This expression does not depend on \(r\), \(\tau\), \(\varepsilon\); it is valid for all elementary tubes into which we mentally divided our cylinder. Consequently, equation (84) determines the ratio of the total electromotive forces \(E_a\) and \(E_b\) induced in the coil for the two types of oscillations considered.

For the ratio of the maximum values of \(E_a\) and \(E_b\), we find

\[ R=\frac{E_{a\max}}{E_{b\max}}=\frac{2Am\omega}{BeH}. \tag{85} \]

§ 52. Details of the Experiments

In the experiments of Tolman, Karrer, and Guernsey the coil was connected (through a tube amplifier and transformer) to a vibration galvanometer tuned to the frequency of oscillation of the cylinder, and the value of \(R\) (and, consequently, also of \(\frac{m}{e}\)) was determined from the ratio of the amplitudes of the galvanometer in the two cases. The sign of \(e\) was not determined. In the experiments of Tolman and Mott-Smith both the magnitude and the sign of the ratio \(\frac{m}{e}\) were determined.

In their last work, an earth inductor, connected (by means of slip rings and brushes) in series with a rheochord provided with two sliding contacts, rotated synchronously with the cylinder during both operations described above. In each of the operations, one end of the coil was connected to one of the sliding contacts of the rheochord, while the other end of the coil and the second contact of the rheochord were connected to the input terminals of the amplifier by means of a special device.

It was possible, arbitrarily and continuously, to vary the phase of the alternating potential difference produced by the inductor along the rheochord relative to the potential difference induced in the coil during the oscillations of the cylinder. In a similar manner, the magnitude of the potential difference between the contacts could be varied at will by simply moving them. In each operation, by adjusting the magnitude of the voltage and the phase, it was possible to obtain zero deflection

of the galvanometer. The length of the wire (between contacts) at equilibrium was proportional to \(E_a\) and \(E_b\). In addition, the phase difference was known between the electromotive force of the terrestrial inductor and the electromotive force induced by the oscillating cylinder during the second operation. Thus, knowing the phase of \(\dfrac{dV}{dt}\) (i.e., of the acceleration of the oscillating cylinder in the first operation) relative to the electromotive force of the terrestrial inductor, it was possible to determine also the phase relation of \(E_a\) and \(-\dfrac{dV}{dt}\).

§ 53. Systematic errors and their elimination

All systematic errors that were foreseen by the authors were carefully studied by them, and it was shown that their influence is very small. A number of errors could arise from the presence of the earth’s magnetic field. Most of these disappeared if the axes of the oscillating cylinder and of the coil surrounding it were strictly parallel to the direction of the earth’s field. It was shown that possible deviations from parallelism, caused by imperfection of the construction, vibrations of the coil support, imperfection of the cylinder bearings, and asymmetry of its oscillations, have only a negligible or very small effect. Expansion of the cylinder, caused by centrifugal forces, could give rise only to a very small electromotive force (about 5% of the measured e.m.f.), whose frequency was twice as high as the frequency of oscillation of the cylinder and which did not affect the galvanometer readings. Phase relations determined by the construction of the instrument and calculated according to a well-developed theory, as well as the purely harmonic character of the cylinder oscillations, were checked optically.

Special experiments with grounding showed that accidental electric charges have no effect. Special precautions were taken to exclude the influence of external electric circuits located near the apparatus. Further, in order to study eddy currents, measurements were made with a solid cylinder assembled from copper rings separated by thin Bakelite rings, but the results of this experiment proved rather unreliable. The authors suppose that, because of the interaction arising between the rings, the electromotive force increases instead of decreasing. They think that, by proceeding along this path, it will probably be possible to explain the increase in the values of \(\dfrac{m}{e}\) observed in the experiments of Tolman and Stewart (§ 50).

§ 54. Results

The measurements of Tolman, Kerrer, and Guernsey gave for the mean value of \(\dfrac{m}{e}\) a value 8% smaller than its standard value...

for free electrons. More refined measurements by Tolman and Mott-Smith gave a value 19% smaller than the standard one; Tolman and Stewart’s measurements, however, gave a value exceeding the standard by 15%. In the work of Tolman and Mott-Smith it turned out that the phase of the electromotive force due to the inertia of the electrons lags by \(10^\circ\) behind the phase \(\left(-\dfrac{dV}{dt}\right)\), whereas the theory requires coincidence of the phases of these quantities. These discrepancies have not yet been explained.

E. Barnett’s experiments with the inertia of electrons\(^{46}\)

§ 55. Experimental methods (§§ 27 and 41)

These experiments were based on methods almost exactly coinciding with the method used by the author in the experiments on rotation under magnetization, when the magnetizing coil was wound directly on the rotor.

The magnetic cylinder was replaced by a brass or glass one; the latter is better, since it makes it possible to exclude the possible influence of the inertia of the electrons in the cylinder. The principal experiment was carried out with a copper coil wound on glass; the entire rotor had the same length as most of the magnetic rotors, but a somewhat larger diameter than the diameter of the largest of the magnetic rotors.

§ 56. Theory of the null method

If the oscillations are eliminated, then, as in § 28, we obtain

\[ -\rho = \Gamma \gamma m_0 X_0, \tag{86} \]

where \(\rho\) is the ratio of the torque of the free charges present in the coil to the magnetic moment of the coil itself. As was shown in § 5, for free electrons rotating in circular orbits this ratio is equal to \(2\dfrac{m}{e}\).

In addition, disturbing moments may arise, either lying in quadrature with the moment due to the inertia of the electrons, or coinciding with it in phase (or differing by \(180^\circ\)). In the first case the amplitude can be reduced to a minimum, but not to zero, by changing the conductivity \(X\). In the second case it can be made equal to zero, but upon substituting the value \(X_0\) into equation (86) one obtains not \(\rho\), but \(\rho + \Delta \rho\), where \(\Delta \rho\) is the error of measurement.

If one investigates the dependence of the amplitude \(A\) on \(X\) for a whole series of values on both sides of \(X_0\), and calculates negative values for one side and positive values for the other (since the torque changes sign when \(X = X_0\)), and then plots \(A\) as a function of \(X\), it is easy to verify that the result should be a straight line, provided that there is no

disturbing moments that are in quadrature with \(t\), and the influence of the amplitude, considered in the following paragraph, may be neglected. If, however, the moment is in phase with \(t\), then the relation must hold

\[ -\rho + \Delta \rho = \Gamma \gamma m_0 X_0, \tag{87} \]

where \(\Delta \rho\), as indicated above, determines the final error in the determination of \(\rho\). If the experiment can be repeated with the opposite phase of the disturbing moment, then there must be

\[ -\rho - \Delta \rho = \Gamma \gamma m_0 X_0', \tag{88} \]

where \(X_0'\) is the new value of the conductivity corresponding to zero amplitude. Then

\[ -\rho = \Gamma \gamma m_0 \frac{X_0 + X_0'}{2}. \tag{89} \]

If, however, there is a moment that is in quadrature with \(t\), then a minimum of amplitude will be obtained, but not zero, as has already been indicated above, and the line will be curved on both sides of the minimum.

If a multilayer coil with a winding is used, and not an induction solenoid, then, as is easy to show, the constant \(\gamma\) must be replaced by the constant \(\gamma' = 4\pi \dfrac{Z_2}{l}\), where \(Z_2\) is the total number of turns of the winding, and \(l\) is the axial length of the homogeneous winding of the rotor.

§ 57. General theory

To develop a more general theory, we shall reason as follows: let the rotor coil contain \(Z\) turns of thin wire of cross-section \(s\) and have mean radius \(r\); let it be traversed by the current

\[ i = I \sin \omega t. \tag{90} \]

We shall denote the number of free electrons per unit volume by \(n\), the angular amplitude of the rotor’s oscillations by \(\theta\), and its angular velocity by \(\dot{\theta}\) (it coincides in phase with the rotating moment \(J\)); the latter is determined by the equation

\[ \dot{\theta} = \omega \Theta \sin(\omega t + \beta). \tag{91} \]

If \(v\) denotes the velocity of an electron, and \(V\) the velocity of the wire, then the current in the coil will be equal to \(i = nea(v - V)\), whence

\[ v = \frac{i}{nea} + V = \frac{i}{nea} + r\dot{\theta} \tag{92} \]

and

\[ v = \frac{i}{nea} + r\dot{\theta}. \tag{93} \]

Then the torque acting on an electron and computed relative to the axis of symmetry of the coil will be equal to

\[ rm\dot v=\frac{rmi}{nea}+r^2m\ddot\theta, \tag{94} \]

and the torque acting on all the electrons situated in \(Z\) turns will be \(2\pi ranZ\) times greater. But since the total torque acting on the rotor is equal in magnitude and opposite in sign to the torque acting on the electrons, we have

\[ t=-\frac{2m}{e}\pi r^2Zi-\frac{2m}{e}\pi r^2Znear\ddot\theta . \tag{95} \]

Using (90) and (91), we can rewrite this equation in the following form:

\[ t=-\frac{2m}{e}\pi r^2ZI\omega\cos\omega t-\frac{2m}{e}\pi r^2Znear\omega^2\Theta\cos(\omega t+\varphi)= \]

\[ =T_1\cos\omega t+T_2\cos(\omega t+\varphi). \tag{96} \]

With the null method, \(T_2\) vanishes, and from (96) one obtains the result found earlier (86).

In these experiments the ratio \(\dfrac{T_2}{T_1}\), almost equal to \(\dfrac{\omega\theta}{I}\), was very small for the amplitudes obtained in the inertial effect. Thus, if one takes \(n=10^{23}\), \(e=5\cdot10^{-5}\), \(a=0.27\), \(\omega=2\pi\cdot14.6\), \(I=0.01\) (all in the CGSM system) and uses the value of \(\theta\) indicated above, the ratio is found to be \(5\cdot10^{-3}\). For the largest amplitudes used in the experiments (in calibration measurements by the deflection method), which exceeded the amplitudes produced under the action of the inertial effect by approximately 150 times, this ratio was of the order of unity.

If \(T_2\) or \(\theta\) does not vanish, then, as can be shown, the expression

\[ \varphi=\operatorname{arc\,cot}\frac{T_2}{T_1} \tag{97} \]

is satisfied with considerable approximation, so that \(t_2\) and \(t_1\) turn out to be almost in quadrature.

For small amplitudes \(T_2\) is negligibly small in comparison with \(T_1\); in this case, instead of the null method, one may use the deflection method, as in the case of gyromagnetic experiments on rotation under magnetization.

§ 58. Results

Three series of measurements by means of the null method (§ 30, A), carried out under very favorable conditions, gave the value

— \(10^7\rho = 1.10 \pm 0.03\); the standard value is \(10^7\rho = 1.13\). Three series of observations by means of the deflection method (§ 30, B) gave the value \((10^7\rho)=0.87 \pm 0.05\). The latter result should be regarded as less reliable than the preceding one.

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