This paper discusses the solution of the vector Laplace's equation using the method of separation of variables to determine the magnetic field of three systems consisting of infinitesimally thin prolate spheroidal current bands and a ferromagnetic prolate spheroidal shell. Two configurations consist of a ferromagnetic spheroidal shell with an internal or external current band. The third configuration consists of a spheroidal shell with both internal and external current bands. The solutions are limited to de currents and linear shell materials. They provide a method for calculating the interaction between the ferromagnetic material and the current band for symmetric and nonsymmetric current bands and a prolate spheroidal shell.
Coefficients for the prolate spheroidal solution of Laplace’s equation are derived in terms of a line of magnetization extending between the focal points of a spheroid. The present paper applies Havelock’s formula to obtain the magnetization corresponding to terms of a prolate spheroidal inverse model. Knowledge of the relationship of the prolate spheroidal coefficients to the equivalent magnetization distribution is required when mathematically modeling distinctly nonspherical static magnetic sources in the near field.
The activities and mean free bonding energies of Na, K, and Ca on alfalfa, soybean, red top, and Reed canary grass roots were determined by the use of the clay membrane electrode technique. Samples of electrodialyzed roots were saturated with each cation by addition as the hydroxide after the exchange capacity of the roots had been previously determined.Examination of the techniques employed showed that reproducible determinations of the activities could be made after a five minute contact of the clay membrane electrode with the root system. Identical readings from both small and large electrodes indicated that there was no problem in obtaining a representative cross section of the root systems for measurement. There was no appreciable liquid junction potential at the point of contact of KC1 bridge with the root system. The cations added were found to be practically all recoverable in a 0.1N HC1 extraction without serious contamination from ions within the roots.It was possible to determine a value for the cation exchange capacity of the roots by titrating the electrodialyzed roots with base, stirring, and maintaining the system at pH 7 for ten minutes. Bonding energy determinations on these systems indicate that K is bonded stronger than Na on alfalfa roots, but Na is bonded stronger than K on soybean roots. The legume roots held all of the cations with greater energy than did those of the grasses; however, there was a shift in the relative values obtained for the grass root systems when compared to those of the legumes.
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