Salt solutions with two or more salts generate ion currents analogous to magnetic field lines

A gradient of a single salt in a solution generates an electric field but not a current. Recent theoretical work by one of us [P. B. Warren, Phys. Rev. Lett. 124, 248004 (2020)] showed that the Nernst-Planck equations imply that crossed gradients of two or more different salts generate ion currents. These currents in solution have associated nonlocal electric fields. Particle motion driven by these nonlocal fields has recently been observed in experiment by Williams et al. [Phys. Rev. Fluids 9, 014201 (2024)], a phenomenon which was dubbed action-at-a-distance diffusiophoresis. Here we use a magnetostatic analogy to show that in the far-field limit, these nonlocal currents and electric fields both have the functional form of the magnetic field of a magnetic dipole, decaying as ${r}^{\ensuremath{-}d}$ in $d=2$ and $d=3$ dimensions. These long-ranged electric fields are generated entirely within solutions and have potential practical applications since they can drive both electrophoretic motion of particles and electro-osmotic flows. The magnetostatic analogy also allows us to import tools and ideas from classical electromagnetism into the study of multicomponent salt solutions.