The easiest way to demonstrate the effect of the observer's motion is through the theory of relativity. Thus if H is the mag netic field as viewed from one system of axes, the magnetic field H' as viewed in a system of axes moving relatively to the first system with velocity v parallel to the axis of x is such that Inserting appropriate values for c, w and a, we readily find E= 2 X 1 e.s.u., i.e., volts per cm.
A solid sphere of negative electricity would, by its rotation give a magnetic field roughly similar to that of the earth, as viewed by an observer who did not participate in the motion. As viewed by an observer on the sphere, however, the considerations referred to above cause a reversal of the apparent horizontal component while leaving the vertical component unchanged. The latter con sideration may be avoided by coating the surface of the sphere with a uniform charge of sign opposite to but of total amount equal to that of the volume distribution, so as to annul the external The magnetic potentials due to the combined distribution is then where, in putting (3= i we have neglected only quantities of the second order in v/c. Thus, the only effect of the observer's mo tion is on the horizontal component H„'.
The quantity of positive and negative electricity in the earth is enormous. Thus if all the positive and negative electricity in a cubic centimetre of the earth's substance were separated and con centrated at two points a centimetre apart, they would attract each other with a force of tons.
If we could annul the positive charge and leave only the nega tive, the rotation of the earth would produce a field about 5 X times that of the earth. Since two spheres of equal charge but different radii produce unequal magnetic fields at a point external to both it occurred to Sutherland that only a slight difference would be necessary in the radii of two superposed spheres of positive and negative, of which the earth may be regarded as being composed in order that, by the combined rotation of these spheres, a field comparable with that of the earth would result. As a matter of fact calculation shows that the radii need only differ by 2.4X cm., i.e., about one hundredth of the diameter of a single atom in order to produce the desired effect. Moreover, since a symmetrical spherical distribution of charge produces at external points an electric field which is independent of the size of the sphere, the two equal and opposite charges would together produce no electric field at an external point. However, as regards regions just inside the surface the condition is different. Thus, at a point
just inside the thin shell of thickness cm., there would be a field of the order 6X id' volts per centimetre tending to annul the separation which had taken place. This is demonstrated thus: The magnetic potential 12 due to a sphere of radius a and uni formly distributed volume charge of total amount Q is, at an external point with co-ordinates r, Q, given by The potential 3f.2 due to two superposed spheres of equal charge, but of radii a and a+ba respectively, is consequently The first term on the right hand side corresponds to the volume charge and the second to the surface charge. The effect of the surface charge is thus to reverse the sign of fl from that given by the volume charge. In other words, in order to produce a mag netic field corresponding to that of the earth we should have to postulate a positive volume charge, and a negative surface charge. Here again, however, the electrostatic pull on the negative surface charge as a result of the volume charge would be enormous, the field just inside the surface layer being of the order volts per centimeter. There is consequently no obvious way in which the separation of charges could be maintained.
A modification of the foregoing possibilities is to be found in supposing that the rotation brings about a radial polarization in the atoms. The rotation of such a polarized system gives rise to a magnetic field, and also, of course, to an electric field. The latter may be compensated at external points by a suitable distribution of charge over the surface, and the rotation of the combined electrostatic distribution would produce a magnetic On submitting the matter to calculation, we find that the polarization necessary to give rise to a magnetic field comparable with that of the earth is such as would correspond to each having the moment which it would acquire by the separation of a proton and electron to the extent of an atomic diameter. (See ATOM.) While the dis tribution of charge over the surface would come about automati cally in the case of conducting atmosphere, it would, however, not remain on the surface since the polarized distribution of the interior would give rise to an enormous field of the order of le volts per centimetre tending to draw it inwards. A possibility closely resembling the foregoing is one suggested by Sir Joseph Larmorm, who examines the consequences of supposing that the earth may be regarded as a rotating polarized crystal.