Inversion Applied to the Solution of 3-Dimensional Electromagnetic Problems

Abstract

This paper presents an extension of the method of inversion to certain electromagnetic problems and rectifies at the same time an error of long standing in the solution of an elementary problem which seems to have acted as a bar to just this extension. If the image of a charge in a sphere is derived by the method of inversion, as is done in this paper, it becomes immediately obvious that the image of radially oriented dipole is another radial dipole plus a free charge proportional to the moment of the dipole. The potential caused by the free charge is not of negligible magnitude, and this indicates that the solution given by Sir J. Larmor and independently by S. P. Thompson and Miles Walker for the case of a magnetic dipole outside a sphere of infinite permittivity cannot be correct, for the image used therein consists of a simple dipole. However, it can be shown in addition that the latter image is not void of physical significance: the vector potential of such an image transforms in a simple manner and from this in turn follow simple laws of transformation for flux linked with loops and for mutual inductance between such loops. This extension of the method of inversion is used in the second part of the paper to deal with a problem complementary to that of Larmor-Thompson, namely, with that of a spherical wall impermeable to magnetic flux. This problem is of practical importance as an idealization of an inductor used at radiofrequencies and placed inside a metal screen, and it has for this reason been dealt with by direct attack on several previous occasions. It is shown here that the action of such a screen on the field produced by the loop can be replaced by the introduction of a simple image loop. From this follows then a simple way of calculating the change in self and mutual inductance produced by such a screen, even in cases more general than considered heretofore. The losses caused by the screen can also be easily calculated by the use of H. A. Wheeler's concept of equivalent magnetic skin thickness.

Quasi-static potential created by an oscillating dipole in the vicinity of two nanospheres (dimer): inversion transformation method

“…We mention that there are other studies in which the inversion transformation was applied for solving electromagnetic problems [25][26][27][28]. Nearly seven decades ago, Bloch applied this method of transformation for analyzing the magnetic fields created by a magnetic dipole outside of a sphere with infinite permittivity and by current loops inside a metal screen [25]; moreover, he finds that a radial dipole is transformed into another radial dipole plus a charge. Almost a quarter of a century ago, by using this transformation, Lin and Jin [26] found the electric potential generated by a point charge in the vicinity of two dielectric spheres.…”