The currents induced inside cells by external electric fields in the frequency range 50–60 Hz are studied analytically by accounting for thin cell membranes with transverse conductivity that is small compared to the conductivity of the saline fluid. A general perturbation scheme is formulated and applied to two adjacent spherical cells of equal radii by using a reflection principle and solving a nonlinear difference equation. The presence of the second cell is found to cause a no more than 10% increase to the current induced in an isolated spherical cell.
The term “interior” in the case with a membrane of finite thickness defines the region occupied by the cell with the exclusion of the membrane. In the limit of vanishingly small membrane thickness the interior does not include the boundary; the rest of space, including the boundary, is the “exterior” of the cell.
As mentioned in the Introduction, elongated cells may be excluded from our analysis.
The issue of convergence of this expansion is not addressed in this paper. It is expected that the expansion converges for sufficiently small .
The perturbation order (subscript in expansion terms) should not be confused with the index in , and other symbols of this section; specifies individual cells.
as means that if for all .
The reason for considering two symmetrically located dipoles instead of one is the ensuing convenience of eliminating any residual constant for the potential in the limit . Specifically, remains zero at the origin for any distance .