Electrostatics of quadrupolarizable media

TL;DR

This paper extends classical electrostatics by incorporating quadrupolarization effects, resulting in a fourth-order Poisson-like equation that significantly alters the behavior of electrostatic fields in media with molecules possessing large quadrupole moments.

Contribution

It introduces a quadrupolar analogue of Poisson's equation, accounting for molecular quadrupole moments and deriving new boundary conditions for electrostatics.

Findings

Quadrupolar effects lead to a fourth-order differential equation for electrostatics.

Additional boundary conditions are required due to quadrupolarization.

Significant differences from classical theory are observed when quadrupole moments are considered.

Abstract

The classical macroscopic Maxwell equations are approximated. They are a corollary of the multipole expansion of the local electrostatic potential up to dipolar terms. But quadrupolarization of the medium should not be neglected if the molecules which build up the medium possess large quadrupole moment or do not have any dipole moment. If we include the quadrupolar terms in Maxwell equations we obtain the quadrupolar analogue of Poisson's equation: $\nabla^2 \phi - L^2_Q\nabla^4 \phi = - \rho / \varepsilon$. This equation is of the fourth order and it requires not only the two classical boundary conditions but also two additional ones: continuous electric field and the relation of the jump of the normal quadrupolarizability at the surface to the intrinsic normal surface dipole moment. The account of the quadrupole moment of the molecules leads to significant differences compared to the classical electrostatic theory.