Electric Flux Approach for Surface Charges on Current Carrying Conductor

Abstract

When a straight conductor of finite length is connected to a battery, the positive and negative charges on the upper and lower plates of the conductor induce an electric field and a current in the conductor. In the conventional method, it is assumed that the internal electric field is uniform as steady current flows in the conductor. However, the electric field is induced by electric charges but not by currents. A steady current can flow in a wire when guided by charge accumulated on the surface of the conductor. Although the existence of surface charge has been confirmed in several studies, an analytical approach to determine the surface charge distribution was successful only when the conductor was considered infinitely long. Numerical methods were used to compute the electric flux incident on the conductor surface and to determine the surface charge that balances the incoming electric flux. However, the relationship between the incoming electric flux and the surface charge has not been established in a closed form. In this study, the conduction current is considered to be the relaxation of the electrode and surface charges and is treated as the sum of the electric fluxes entering and leaving the conductor surface. The relationship between the incoming electric flux and surface charge is obtained in a closed form by applying the continuity equation and boundary conditions for the electric field at the interface between two dissimilar materials. This relationship is applicable to any interface, including conductor-conductor and lossy dielectric-dielectric interfaces. A two-dimensional model of a current-carrying wire is developed using slab conductors to numerically compute the surface charge densities, electric fields, and equipotential surfaces inside and outside the conductor. Our results show that the charge peaks at the corners of the conductor and at the point where two dissimilar conductors meet.