Some problems of the relation between disturbed magnetic force and a magnetized body
Yumura, T.
Abstract
In his previous papers, the writer gave the results of distribution and the local anomalies of the Earth's magnetic field, but in attempting to discuss the anomalies theoretically, he had to put some assumptions to the following factors:
(1) the geometrical form and the position of the vertical direction of the subterranean body,
(2) the cause of the magnetization of the body,
(3) the direction and the magnitude of the magnetization.
At first, for (1), the form and depth shall be roughly imagined from the disturbed field on the Earth's surface and geological structure, excepting the case when they were found by the boring.
In either case, however, the ideal form must be considered for theoretical discussion.
Next, as regards the second, we may be able to consider the following causes:
(1) the Earth's magnetic induction due to the substance of the earth's crust,
(2) the permanent magnentism of rocks,
(3) the electromagnetic action due to a local earth-current.
But we can't easily determine to which of the three the cause of magnetization belongs.
In general, theoretical calculations will be carried out under the condition that the magnetization is caused by the Earth's magnetic induction, and uniform all over the body, with its direction parallel to the Earth's magnetic field, having the magnitude defined by the susceptibility and the field. In practice, those idealised conditions above mentioned are not fulfilled.
If a substance of an arbitrary form is placed in the Earth's magnetic field(generally, in a uniform field), it is not uniformly magnetized; that is, the magnetic induction at every point is not the same.
The form of the substance, in which it is uniformly magnetized, is only an ellipsoid. Even if the substance is uniformly magnetized, the direction of its magnetization is not parallel to that of the outer field. The form of the substance, in which the direction of the magnetization is parallel to that of the outer field, is only a sphere.
If the magnetization of the body depends only on magnetic induction, A, B, C-x, y, z-components of the magnetization respectively - are defined by
A = Fox / (1/κ + Nx), B = Foy/(1/κ + Ny), C = Foz / (1/κ + Nz)
where Fo is the outer field, κ the susceptibility, and N the demagnetizing factor.
In general these components of the demagnetizing factor are not equal, and the direction of magnetization, usually, is not parallel to the outer field. Only in the case of which κ is smaller than 10-3, the contribution of the factor to the direction of the magnetization becomes out of consideration.
Under such assumptions, the writer gives, in this paper, the field of the rectangular solid placed in a uniform field.
The writer has mentioned in this paper that the direction of the magnetization can not be determined so easily, when the body bas a permanent magnetism, or is magnetized by induction. The fol1owing method, however, will bring to light this problem to certain degree.
If the origin of a rectangular coordinates is taken appropriately in or near the body, such as r>ro, where r is the distance from the origin to some external point and ro from the origin to any volume element in this body, then the distance from the volume element to the external point is represented by a spherical harmonic series.
Hence, if the magnetization of this body is uniform, the magnetic potential, consequently the magnetic force, due to this body is represented by a spherical harmonic series; that is
X = ΣSn1∫ronPndv Y = ΣSn2∫ronPndv Y = ΣSn3∫ronPndv
Since Sn1,Sn2,Sn3 are linear functions of A,B,C, the values of APn', BPn', CPn' are obtained (Pn'∫ronPndv). Hence, by taking the ratio of these values, the direction of the magnetization will be found.
