Forces in Nonlinear Media

TL;DR

This paper explores forces in nonlinear media governed by a generalized Poisson equation, deriving force expressions for point charges and bodies, and analyzing effects of different coupling constants on interactions.

Contribution

It provides exact force formulas for point charges in nonlinear media and generalizes classical force laws to complex, nonlinear systems with variable response coefficients.

Findings

Force on a point charge at low and high limits derived

Force on a body in a constant gradient field is proportional to total effective charge

Sign of G determines attraction or repulsion between charges

Abstract

I investigate the properties of forces on bodies in theories governed by the generalized Poisson equation div[mu(abs(grad_phi))grad_phi]=G rho, for the potential phi produced by a distribution of sources rho. This equation describes, inter alia, media with a response coefficient, mu, that depends on the field strength, such as in nonlinear, dielectric, or diamagnetic, media; nonlinear transport problems with field-strength dependent conductivity or diffusion coefficient; nonlinear electrostatics, as in the Born-Infeld theory; certain stationary potential flows in compressible fluids, in which case the forces act on sources or obstacles in the flow. The expressions for the force on a point charge is derived exactly for the limits of very low and very high charge. The force on an arbitrary body in an external field of asymptotically constant gradient, E, is shown to be F=QE, where Q is the total effective charge of the body. The corollary Q=0 implies F=0 is a generalization of d'Aembert's paradox. I show that for G>0 (as in Newtonian gravity) two point charges of the same (opposite) sign still attract (repel) each other. The opposite is true for G<0. I discuss the generalization of this to extended bodies, and derive virial relations.