Debye-H\"uckel potential at an interface between two media

Electrostatic interactions between point charges embedded into interfaces separating dielectric media are omnipresent in soft matter systems and often control their stability. Such interactions are typically complicated and do not resemble their bulk counterparts. For instance, the electrostatic potential of a point charge at an air-water interface falls off as $r^{-3}$, where $r$ is the distance from the charge, exhibiting a dipolar behaviour. This behaviour is often assumed to be generic, and is widely referred to when interpreting experimental results. Here we explicitly calculate the in-plane potential of a point charge at an interface between two electrolyte solutions with different dielectric permittivities and Debye screening lengths. We show that the asymptotic behaviour of this potential is neither a dipole, which characterises the potential at air-water interfaces, nor a screened monopole, which describes the bulk behaviour in a single electrolyte solution. By considering the same problem in arbitrary dimensions, we find that the physics behind this difference can be traced to the asymmetric propagation of the interaction in the two media. Our results are relevant, for instance, to understand the physics of charged colloidal particles trapped at oil-water interfaces.