Gravitational-Plasma Hypothesis of Terrestrial Magnetism and the Earth's Charge
Abstract Generated abstract
The paper proposes a gravitational-plasma hypothesis for terrestrial magnetism in which centrifugal effects in a rotating, high-pressure plasma core separate heavy positive ions from electrons, producing moving charge distributions that generate magnetic fields. It develops an approximate model of these currents, first as equatorial loops and then as a surface distribution on the outer boundary of the core, using vector magnetic potential calculations with elliptic integrals and numerical computation. The authors estimate the surface charge density required to match the observed equatorial magnetic-field intensity and infer a large positive charge on the core boundary, then suggest that the Earth’s observed negative surface charge could arise from ionospheric electrons attracted by this internal positive charge.
Full Text
GRAVITATIONAL–PLASMA HYPOTHESIS OF TERRESTRIAL MAGNETISM AND OF THE EARTH’S CHARGE
A. A. VOROB'EV, T. Yu. MOGILEVSKAYA
It is assumed that the Earth’s core \(E\) is filled with plasma under high pressure. In the plasma zone \(E\), as the Earth rotates, charge separation occurs under the action of centrifugal force: the heavy, positively charged remnants of atoms move farther away from the axis of rotation than the light electrons. The separation continues until the action of the centrifugal force is balanced by the Coulomb attraction between unlike charges. In this state of dynamic equilibrium, the total positive charge located on the surface of zone \(E\) will be equal to the negative charge of the electrons remaining in the inner layers of this zone. Both charges, moving in space together with the Earth’s core as it rotates about its axis, form electric currents, each of which creates its own magnetic field. The Earth’s magnetic field may be regarded approximately as the result of the superposition of these fields.
Since, in accordance with existing theories, the rotational velocities of the geospheres differ, zone \(E\) rotates about the axis with a velocity different from the rotational velocity of the Earth’s crust and from the velocity of an observer located on the Earth’s surface. Consequently, according to the principle of relativity, the Earth’s magnetic field created by the motion of the charges of zone \(E\) can be detected and measured on the Earth’s surface [1].
The calculation according to this hypothesis is given below.
If the motion of the charges separated in the manner indicated is represented in the form of circular currents concentrated in the equatorial region of zone \(E\), then the total flux is
\[ \Phi=(L_1-L_2)i, \tag{1} \]
where \(L_1\) and \(L_2\) are the inductances, respectively, of the inner and outer contours (Fig. 1), and \(i\) is the current, which in both contours, as is not difficult to show, is approximately the same.
Taking into account only the external inductance of each contour, we obtain the expression for the magnetic flux in the form
\[ \Phi=\mu_0 i\left[R_2\ln 8R_2-R_1\ln 8R_1-(R_2-R_1)\ln r-2(R_2-R_1)\right], \tag{2} \]
where \(r\) is the radius of the cross section of the contour;
\[ R_1 \approx 1400\ \text{km} \]
is the inner radius of zone \(E\);
\[ R_2 \approx 350\ \text{km} \]
is the outer radius of this zone.
In numerical values,
\[ \Phi=(42-6.08\lg r)i\ \text{Wb}. \]
The value of the total flux is positive even for very large cross sections of the current loops. Thus, for \(r=1000\ \text{m}\), \(\Phi=23.76\, i\) Wb; for \(r=1000\ \text{km}\), \(\Phi=5.52\, i\) Wb.
The direction of the total magnetic flux can be determined with the aid of Fig. 1. The Earth rotates with angular velocity \(\omega=73\cdot 10^{-6}\ \text{s}^{-1}\) from west to east; the charges separated under the action of the centrifugal force move together with it. The presence of a westward drift indicates that zone \(E\) rotates more slowly than the outer shells of the Earth. Therefore the direction of the relative rotational velocity of zone
Fig. 1. Diagram of the distribution of the Earth’s magnetic field in accordance with the gravitational-plasma theory.
\(E\) and the direction of the current produced by the positive charges will be from east to west. The motion of electrons in the same direction may, in accordance with the adopted positive direction of current, be regarded as a current in the opposite direction. The total magnetic flux \(\Phi\) has a direction within the Earth’s core from the northern geographic pole to the southern one, closing in near-Earth space in the direction observed in nature: from the southern geographic pole to the northern one.
In a more exact calculation of the Earth’s magnetic and electric fields, one should take into account that the positive charges, though nonuniformly, are distributed over the entire surface of zone \(E\), and not only in a thin band along the equator. The negative charges, which experience almost no centrifugal force, are arbitrarily scattered in a certain volume adjacent to the inner boundary of zone \(E\). Therefore the principal role in the creation of the Earth’s magnetic field is played, as indicated above, by the current formed by the motion of the positive charges. Below we shall confine ourselves to calculating the field of this current.
The separation of charges and the concentration of positive charges at the outer boundary of the zone are determined by the centrifugal force
\[ f=\frac{mv^2}{R}=\frac{4\pi^2 mR}{T^2}, \tag{3} \]
where \(v\) is the linear velocity,
\[ R=K_d\cos\alpha \]
is the radius of curvature of the trajectory (Fig. 2),
\[ R_0=3470\ \text{km} \]
is the outer radius of zone \(E\),
\[ T=86400\ \text{s} \]
is the time of one revolution of the Earth.
Assume that the surface charge density \(\sigma\) at any point-
toward the surface is proportional to the value of the centrifugal force. Then, if the charge density at points of the equator is denoted by \(\sigma_0\), we obtain the surface charge density at an arbitrary point as
\[ \sigma=\sigma_0\frac{f}{f_0}=\sigma_0\frac{R}{R_0}=\sigma_0\cos\alpha. \tag{4} \]
Fig. 2. Schematic section of the Earth (for calculating the vector magnetic potential).
The entire current formed by the motion of these charges may be represented as a set of linear circular currents \(di\), each of which is determined by the charge density and the velocity of its motion; moreover, for judging the current that creates the magnetic field which can be observed at the Earth’s surface, one should bear in mind the relative velocity \(v_1\), determined from the data on westward drift. Zone \(E\) lags in its rotation behind the outer layers of the Earth by \(0.02^\circ\) per year. Therefore the relative velocity may be determined as
\[ v_1=\frac{2\pi R\cdot0.02}{360\cdot86400\cdot365} =\frac{v}{6.57\cdot10^6}. \]
The elementary linear current is
\[ di=\sigma v_1\,ds= \frac{2\pi\cdot10^{-6}}{6.57\,T}\,\sigma_0 R_0^2\cos^2\alpha\,d\alpha. \tag{5} \]
The vector magnetic potential \(dA\) at a point \(M\) on the Earth’s surface \((R_3=6370\ \text{km})\) due to a circular current element \(i\) is determined (Fig. 2) as
\[ dA=\frac{\mu_0 R\,di}{4\pi} \int_0^{2\pi} \frac{\cos\beta\,d\beta} {\sqrt{z^2+R^2+R_3^2-2RR_3\cos\beta}}. \tag{6} \]
Using elliptic integrals,
\[ dA=\frac{\mu_0 di}{2\pi} \sqrt{\frac{R_0\cos\alpha}{R_3}}\cdot f(\kappa), \tag{7} \]
where
\[ f(\kappa)=\left(\frac{2}{k}-\kappa\right)K-\frac{2}{k}E, \tag{8} \]
\[ \left. \begin{gathered} \kappa^{2}=\frac{4R_{3}R_{0}\cos\alpha}{z^{2}+(R_{3}+R_{0}\cos\alpha)^{2}} =\frac{4R_{3}R_{0}\cos\alpha}{R_{3}^{2}+R_{0}^{2}+(2R_{3}+R_{0}\cos\alpha)^{2}};\\ z=R_{0}\sin\alpha;\\ K=\int_{0}^{\pi/2}\frac{d\gamma}{\sqrt{1-\kappa^{2}\sin^{2}\gamma}};\\ E=\int_{0}^{\pi/2}\sqrt{1-\kappa^{2}\sin^{2}\gamma}\,d\gamma;\\ \gamma=\frac{1}{2}(\beta-\pi). \end{gathered} \right\} \tag{9} \]
In numbers,
\[ \kappa^{2}=\frac{88.6\cos\alpha}{52.65+44.3\cos\alpha}. \]
The vector magnetic potential at the point \(M\) (Fig. 2) from the total current formed by the positive charges located on the outer surface of the zone \(E\),
\[ A=2\int_{0}^{\pi/2}\frac{\mu_{0}\,di}{2\pi}\sqrt{\frac{R_{0}\cos\alpha}{R_{3}}}\cdot f(\kappa)= \]
\[ =\frac{2\mu_{0}\cdot10^{-6}}{6.57\,T}\sqrt{\frac{R_{0}}{R_{3}}}\cdot R_{0}^{2}\sigma_{0}\int_{0}^{\pi/2}f(\kappa)\cdot(\cos\alpha)^{2.5}\,d\alpha, \tag{10} \]
which, after substitution of the numerical values, is
\[ A=38.3\cdot10^{-6}\sigma_{0}\int_{0}^{\pi/2} f(\kappa)(\cos\alpha)^{2.5}\,d\alpha. \tag{10a} \]
The intensity of the magnetic field on the Earth’s surface at points of the equator is
\[ H_{0}=\left[\frac{1}{\mu_{0}R}\frac{\partial(RA)}{\partial R}\right]_{R=R_{3}}. \tag{11} \]
Expressions (10a) and (11) were transferred to the “Promin” computer. The calculation was carried out on two jointly operating machines, one of which determined the elliptic integrals \(K\) and \(E\), while the second performed the remaining computations.
For an arbitrarily specified value \(\sigma_{0}'=10^{-3}\ \mathrm{C/m^{2}}\), the dependence \(H'(R)\) was computed as \(R\) varied from 6200 to 6600 km in intervals of 20 km, for each of which the mean value \(H'\) was obtained. The dependence \(H'(R)\), plotted graphically, turned out to be almost rectilinear (Fig. 3). For \(\sigma_{0}'=10^{-3}\ \mathrm{C/m^{2}}\), the magnetic-field intensity on the Earth’s surface at points of the equator, \(H_{0}'\), was found to be \(6.35\ \mathrm{A/m}\). According to geophysicists’ data, however, this intensity is \(H_{0}=0.25\ \mathrm{Oe}=19.9\ \mathrm{A/m}\).
Fig. 3. Dependence of \(H^{1}(R)\) for surface charge density \(\sigma_{0}=10^{-3}\ \mathrm{C/m^{2}}\).
To obtain such a field strength, the surface charge density in the equatorial region of zone \(E\) must be
\[ \sigma_{0}=\sigma'_{0}\frac{H_{0}}{H'_{0}}=3.14\cdot 10^{-3}\ \mathrm{C}. \]
With such a surface charge density, the total charge \(q\), distributed over the surface of zone \(E\), will be equal to the enormous quantity:
\[ q=\int_{0}^{\pi}\sigma\cdot 2\pi R R_{0}\,d\alpha =\sigma_{0}\cdot 2\pi R_{0}^{2}\int_{0}^{\pi}\cos^{2}\alpha\,d\alpha =\pi^{2}R_{0}\sigma_{0}=37.4\cdot 10^{10}\ \mathrm{C}, \]
where the charge is positive.
The strength of the electric field at the surface of the Earth, in accordance with Gauss’s theorem, is determined by the total charge of the Earth, which according to [2] is \(5.4\cdot 10^{5}\ \mathrm{C}\), the charge being negative. The vertical component of the electric-field strength at the Earth’s surface varies, according to [3], within the limits \(3\div 100\ \mathrm{V/m}\).
Probably, the presence of the Earth’s negative charge could be explained with the help of the following hypothesis: the total charge of zone \(E\) is close to zero, but the charges of this zone are separated in such a way that a large positive charge is located on its surface. Therefore it seems possible to suppose that electrons arising in the ionosphere owing to absorption of the Sun’s ultraviolet rays are attracted to this positive charge and settle on the Earth’s surface. As a result, the surface charge of the Earth turns out to be negative.
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- N. V. Krasnogorskaya. Investigation of the Electricity of Clouds and Precipitation. Abstract of doctoral dissertation. Moscow, 1968.