The complete inadequacy of non-electrical forces of ordinary type to maintain against the electrostatic forces such charge dis tributations as would be necessary to produce, by their rotation, a magnetic field comparable with that of the earth may be illustrated by one or two examples. Thus, the centrifugal force of the earth's rotation would, in a conducting earth cause a flow of electrons to the exterior until compensation was attained by the electrostatic forces. On submitting the matter to we find that the magnetic field which results is only about of the earth's field. Moreover, for an observer situated on the sphere, the hori zontal component H would be of the same sign as that of the earth for latitudes o° to 26.6°, and of opposite sign for other latitudes, while the vertical component would be of the same sign as that of If the free electrons in the substance of the earth are affected by gravity, we should expect them to tend to congregate towards the centre until equilibrium was established on account of their pressure and electrostatic repulsion, the latter equilibrating in fluence being the all important one. On submitting the matter to calculation"), we find that the magnetic field which results is of the order of the earth's field, and is moreover opposite in sign to that of the earth.
The solution for the foregoing case gives (a) a uniform negative volume charge of such amount as to give rise to a magnetic po tential D being the density of the earth's substance, and G the gravita tional constant. (b) A uniformly distributed positive surface charge of total amount equal to that of the volume charge, and giving rise to a magnetic potential There is of course no external electrostatic field.
Owing to the fact that the inside of the earth is hot, the pres sure of the free electrons in the earth's interior will exceed that in the cooler parts, and there will be a flow of electrons until the remaining difference in pressure is balanced by the difference of potential set up. The potential difference which arises in this manner is none other than that corresponding to the well-known Thomson effect. The rotation of the charge distribution set up by this process gives rise to a magnetic field. If we submit the matter to calculation") on the classical theory of electron equilibrium, we find that the results depend upon the type of the radial temperature and are greatest for the case where the gradient takes place entirely in a shell near the surface. In this case, assigning a total temperature change of 5,000° C, we find that the resulting magnetic field produced is of the order of of the earth's field. The sign of the field corresponds to that of the earth.
The analysis for this case shows a volume distribution of posi tive charge and a surface distribution of negative charge. Since both are symmetrically distributed (although the volume charge is not uniformly distributed) there is no external electric field.
If, in accordance with the customary assumption in electron theory of thermionic effects we assume the number n of free electrons per c.c. to be proportional to the 1.5 power of the abso lute temperature T, we find for the magnetic potential due to the combined system of charges when R is the distance from the centre of the sphere to a place where the temperature is T, and T. and are the temperatures
of the surface and centre of the sphere respectively. For the case where the whole temperature drop occurs in a thin shell, near the surface the integral assumes the form (T .) The foregoing considerations will serve to emphasize the gen eral principle that, at any rate, unless we are willing to alter our fundamental laws, it is impracticable to attempt any explanation of the earth's magnetism as a result of the rotation of charges which have been separated against electrostatic attraction, since the mechanical forces necessary to produce the required separa tion must be, in all cases, enormous and far beyond the limits of such forces as are available.
If, assuming that unit charges of either sign are defined so that the repulsive force between like charges, is also as sume that the force exerted by a positive charge on a negative charge e_ is (I +a) e+2_ and the force exerted by a negative charge e_ on a positive charge is (1— (3) where a and (3 are small positive quantities, we obtain a wider range of pos sibility"'.
It would, on this hypothesis, result that a free electron in a neutral sphere would not be in equilibrium since the attraction on it due to the positive charges within would outweigh the repul sion due to the negative. It would be necessary for the sphere to acquire a negative charge in order that equilibrium should be established, and it is possible to determine a so that this negative charge would be such as to give rise by its rotation to a magnetic field comparable with that of the earth. While (3 plays no part in determining the equilibrium of a free electron, it does play a part in determining the total force which would be exerted on the element of matter composed as it is of both positive and nega tive electrons; and, it is possible to determine (3 so that the net force on the element of matter is just such as to correspond to gravitation. In order to account on these lines for both terrestrial magnetism and gravitation, it is necessary to have and I—g=i —(1.9 X so that both a and are zero to within all limits of detection by laboratory experiments. We obtain by our hypothesis a volume charge sufficient to give by its rotation the earth's magnetic field; but, our modification of the law of attraction has saved us from the consequences of a very large electric field such as we should encounter on classical lines; and, indeed, the residual electrical forces have been adjusted so as to no more than satisfy the de mands of gravitation.