AN ATTEMPT AT AN EXPERIMENTAL VERIFICATION OF BLACKETT’S THEORY OF TERRESTRIAL MAGNETISM

The fundamental law recently formulated by Blackett1, relating the magnetic moment of a rotating massive body to its angular momentum, is based, as is known, on a comparison of the magnetic and angular momenta of the Earth, the Sun, and the star 78 Virginis. However, the data relating to the last two objects contain significant errors and, being connected with certain more or less reliable hypotheses (for example, concerning the distribution of density in the body of the star), do not provide grounds for confident judgments about the validity of the law in question. The most convincing, and apparently the only argument in its favor for the present, is the striking agreement—at least in order of magnitude—of the ratios of the magnetic moments of the three indicated objects to their angular momenta, while the moments themselves differ by a factor of \(10^{10}\).

On the other hand, if this law is valid, it must inevitably entail a number of fundamental consequences affecting our basic physical conceptions, and first of all the development of a theory of a unified gravitational and electromagnetic field. Thus every new possibility of testing this law—far from proved, and rather only conjectured by Blackett—acquires exceptionally great significance in principle. It is precisely from this point of view that one should approach the attempt at its direct experimental verification undertaken by Hales and Gough2.

The experiment which they carried out is of interest above all because it makes it possible to avoid the principal difficulties inherent in previously proposed methods, namely: the impossibility of carrying them out under laboratory conditions on a body possessing a sufficiently large angular momentum, and the inevitability of invoking a number of insufficiently substantiated hypotheses when using data relating to stars or planets.

The idea, which according to the authors belongs to Bullard, is very simple and is based on the elementary fact that different mechanisms for the origin of the Earth’s magnetic field must correspond, in general, to different laws of variation of the intensity of the observed magnetic field as a func-

of the observer’s depth of descent into the interior of the Earth. The authors indicate that the calculation of the horizontal component of the intensity of the Earth’s magnetic field as a function of depth, performed by Runcorn, for the case of the validity of Blackett’s law and for the case of the validity of theories explaining terrestrial magnetism by processes taking place in the central core of the Earth, led to the following expressions:

\[ H_d = H_0 \left\{ 1 - 2 \left( \frac{5 \rho_1}{\rho} + 1 \right) \frac{d}{a} \right\} \]
— for Blackett’s theory

and

\[ H_d = H_0 \left( 1 - 3 \frac{d}{a} \right) \]
— for the central-core theory.

Here \(H_0\) is the magnetic-field intensity at the surface of the Earth, \(H_d\) is the magnetic-field intensity at depth \(d\), \(a\) is the radius of the Earth, \(\rho_1\) is the mean density of the ground in the layer from the surface of the Earth to depth \(d\), and \(\rho\) is the mean density of the Earth.

In particular, for a depth \(d = 1.5\) km one obtains

\[ H_d - H_0 = -26\gamma^{*}) \]
— for Blackett’s theory

and

\[ H_d - H_0 = +11\gamma \]
— for the central-core theory.

Later Chapman\(^3\), on the basis of a more rigorous consideration, arrived at a somewhat different expression for the case of the validity of Blackett’s theory,

\[ H_d = H_0 \left\{ 1 - 3 \left( \frac{5\rho_1}{k\rho} - 1 \right) \frac{d}{a} \right\}, \]

where \(k = \dfrac{I}{I_0}\), \(I\) is the moment of inertia of the Earth and \(I_0\) is the moment of inertia of a sphere of the same dimensions and mass, but of constant density. Taking \(\rho = 5.5\), \(\rho_1 = 2.8\), and \(k = 0.88\), he obtained, for \(d = 1.5\) km:

\[ H_d - H_0 = -21\gamma. \]

It is immediately evident that, although the effect is small, with the observance of known precautions it is quite measurable. Unfortunately, the authors themselves were unable to ensure the necessary accuracy of the measurements, and their results should be regarded as preliminary.

The measurements were made in mines with the aid of a Schmidt horizontal magnetometer. Three times (on different days) data were obtained for five different points, the mean depth of which below the mean surface of the ground was 1.5 km.

Allowance for the diurnal variations of the magnetic field was made with the aid of a similar instrument located on the surface.

The main shortcoming of the measurements is, undoubtedly, the insufficient stability of the instrument readings. As the authors indicate, the readings of the instrument before and after its descent underground differed (with variations taken into account) by 14, 22, and 32 \(\gamma\) for the three separate days.

In order to take account of the error arising from the reading, the authors used two procedures:

1) It was assumed that this difference was the result of inaccuracy in the readings of the instrument, and \(H_d - H_0\) was calculated from \(H_0\) as the mean of the values obtained before and after lowering the instrument into the mine.

2) It was assumed that the discrepancy increases linearly with time.

Both procedures gave almost coincident results.

The means for all fifteen measurements of \(H_d - H_0\) proved to be equal to:

\[ H_d - H_0 = -26 \pm 4\gamma \]
— under assumption (1),

*) Let us recall that \(1\gamma = 10^{-5}\) gauss. The mean magnetic field of the Earth has an intensity of about \(5 \cdot 10^{4}\gamma\). Variations are usually several tens, and sometimes a thousand \(\gamma\).

and

\[ H_d-H_0=-24\pm 4\gamma \]

under assumption (2).

It should be noted at once that such good agreement of the measurement results with Blackett’s theory appears to be only seeming. Before comparing them, it is necessary to take into account the influence of the local geological structure. According to the authors’ estimates, under the conditions of their experiments this influence should manifest itself in a decrease of the absolute value of \((H_d-H_0)\) by a minimum of \(6\gamma\) and a maximum of \(14\gamma\), i.e., for \(H_d-H_0\) one obtains values lying between \(-11\pm 5\gamma\) and \(-19\pm 5\gamma\).

Thus, the change in the intensity of the Earth’s magnetic field with depth actually observed in the experiments argues against theories that explain terrestrial magnetism by processes occurring in the Earth’s central core. On the contrary, the discrepancy between the observed value of \(H_d-H_0\) and the predictions of Blackett’s theory is very small. The authors believe that this discrepancy will diminish if a more careful account is taken of the geological structure of the area of measurement, and also if the following circumstance is taken into account. In Runcorn’s formulae there appears the depth reckoned from the mean level of the surface of the whole Earth. In reality, at the place of observation the mean level of the Earth’s surface, relative to which the depth of immersion of the instrument was measured, rises above sea level by approximately 1600 m. Such a local curvature of the Earth’s surface and the displacement of the point of reference relative to the center of the system should cause some change in the theoretical value of \(H_d-H_0\), and specifically in the direction of decreasing its absolute value.

In summary, one may say that although the measurements of Hales and Gough testify in favor of Blackett’s theory, they still cannot serve as decisive proof of its validity. At the same time, the path they have outlined is apparently the most accessible and promising one. It should be expected that careful measurements carried out at different depths and at various points on the Earth’s surface will make it possible to answer unambiguously the question of the validity of Blackett’s theory.

T. Rosenberg

CITED LITERATURE

1. P. M. S. Blackett, UFN 33, 52 (1947).
2. A. L. Hales and D. I. Gough, Nature 160, 746 (1947).
3. S. Chapman, Nature 161, 52 (1948).