Charge Density in a Current-Carrying Wire

Discussions of the force on a charged particle outside a current-carrying wire often assume that the wire is electrically neutral. This problem explores how this assumption is not quite correct. We give solutions both in the lab frame and in the rest frame of the conduction electrons, using both Maxwell’s equations and special relativity. We note that for current to flow in a resistive wire, there must be an axial electric field inside the wire, which requires a surface charge distribution that varies with position along the wire. See, for example, sec. 17 of [1], and [2, 3]. The surface charge distribution could include a uniform term of any magnitude. These surface charges are kept from leaving the surface by quantum effects often summarized by the term “work function.” Likewise, the positive charges in the interior of the wire are held together in a lattice by quantum effects. We suppose that the positive charge density ρ+ is uniform in the lab frame. In this problem we assume that the conduction electrons can be described classically. Then, for steady axial motion, there must be zero radial force on these electrons. We use a cylindrical coordinate system (r, φ, z) whose axis is the axis of the wire. We suppose that the flow of conducting electrons is purely axial, and azimuthally symmetric. The negative charge density, ρ−, could depend on the radius r. The electric field E has no azimuthal component, while the magnetic field B has only an azimuthal component.