Higher Order Multipoles Research Articles

Multipole representation of an eccentric dipole and an eccentric double-layer

Recently a theorem for representing current generators in a volume conductor by the superposition of a central dipole, quadrupole, octopole, etc., has been established by G. C. K. Yeh, J. Martinek, and H. de Beaumont (Bull. Math. Biophysics,20, 203–16, 1958). This theorem makes possible the representation of any discrete or line, surface- or volume-distributed current source by a unique model which can be determined for each given case by surface potential measurements and closed form analysis. In this paper the multipole representations of an eccentric dipole and an eccentric double-layer are obtained in terms of the various parameters of the assumed singularities, and the contributions to surface potentials due to each of the multipoles are compared. Certain numerical results corresponding to those of E. Frank (Amer. Heart J.,46, 364–78, 1953) are carried out and compared. Furthermore, the multipole representation of a partially damaged double-layer is also determined and compared with that of an undamaged one. It is concluded that within the range of parameters corresponding to human subjects the higher-order multipoles can contribute significantly to the surface potentials compared with the dipole.

The relative permittivity of tetragonal arrays of perfectly conducting thin discs

The methods available for the calculation of the interaction field of an array of dipoles are discussed with reference to an artificial dielectric formed from a tetragonal array of conducting discs. A modified method is described and used to derive a formula for the relative permittivity, valid when the discs are not too closely spaced. For closely packed discs, the fields due to higher-order multipoles become important and a second formula for the relative permittivity is obtained for this case. Comparison with experimental values already obtained shows that the two formulae together enable the calculation of the relative permittivity over the whole range of spacings likely to be used in practice to be made.