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We consider two long resistive straight parallel wires carrying opposite constant currents and calculate the potential and electric field everywhere in space and the surface charge densities along the wires. The problem is solved through Laplace's equation in bi-cylinder coordinates, far from the battery. We compare these calculations with previous known results that used different methods. We also calculate the numerical solution for the case in which the battery is present, and show the equipotentials and surface charges close to the battery.

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In this paper we review applications given by J H Poynting (1884) on the transfer of electromagnetic energy in DC circuits. These examples were strongly criticized by O Heaviside (1887). Heaviside stated that Poynting had a misconception about the nature of the electric field in the vicinity of a wire through which a current flows. The historical review of this conflict and its resolution based on the consideration of electrical charges on the surface of the wires can be useful for student courses on electromagnetism or circuit theory.

The question of the charge neutrality of a conductor, carrying a steady electric current, is examined. It is argued that apart from the well-known surface charge distribution (needed for a steady current along the conductor wire length) and the radial non-uniformities arising from some dynamical effects, there is no other net charge distribution, contrary to what has recently been claimed to arise from some kinematic relativistic effects, along the conductor wire in a closed current loop, as seen in the lab frame. On leave from NCRA, Tata Institute of Fundamental Research, Poona University Campus, 411 007 Pune, India.

We present the opinion of some authors who believe there is no force between a stationary charge and a stationary resistive wire carrying a constant current. We show that this force is different from zero and present its main components: the force due to the charges induced in the wire by the test charge and a force proportional to the current in the resistive wire. We also discuss briefly a component of the force proportional to the square of the current which should exist according to some models and another component due to the acceleration of the conduction electrons in a curved wire carrying a dc current (centripetal acceleration). Finally, we analyze experiments showing the existence of the electric field proportional to the current in resistive wires.

We consider a two-wire resistive transmission line carrying a constant current. We calculate the potential and electric eld outside the wires showing that they are diierent from zero even for stationary wires carrying dc currents. We also calculate the surface charges giving rise to these elds and compare the magnetic force between the wires with the electric force between them. Finally we compare our calculations with Jefimenko's experiment.

We treat the problem of two resistive plates carrying a steady current in the same direction. We consider a linear battery orthogonal to the direction of the current in the middle of the plates. We study the behavior of the surface charges close to the battery. We calculate the potential and electric field in the space outside the plates. We also consider the case of a single resistive plate carrying a steady current.

In this work we treat a resistive toroidal conductor carrying a steady azimuthal current. We calculate the electric potential everywhere in space. We also present the electric field inside and outside the toroid and the surface charge distribution along the conductor. We compare our theoretical result with Jefimenko's experiment.

Let us first state exactly what this book is and what it is not. It is a compendium of equations for the physicist and the engineer working with electrostatics, magne tostatics, electric currents, electromagnetic fields, heat flow, gravitation, diffusion, optics, or acoustics. It tabulates the properties of 40 coordinate systems, states the Laplace and Helmholtz equations in each coordinate system, and gives the separation equations and their solutions. But it is not a textbook and it does not cover relativistic and quantum phenomena. The history of classical physics may be regarded as an interplay between two ideas, the concept of action-at-a-distance and the concept of a field. Newton's equation of universal gravitation, for instance, implies action-at-a-distance. The same form of equation was employed by COULOMB to express the force between charged particles. AMPERE and GAUSS extended this idea to the phenomenological action between currents. In 1867, LUDVIG LORENZ formulated electrodynamics as retarded action-at-a-distance. At almost the same time, MAXWELL presented the alternative formulation in terms of fields. In most cases, the field approach has shown itself to be the more powerful.

In order to constrain electrons to move along ohmic conductors carrying steady currents, there must be a surface charge density that is usually very difficult to calculate. An approximate analytic expression for this surface charge density on a conducting square ring is presented here where the only source of emf is a changing external magnetic field. The corresponding electric field is determined and it is checked that the energy balance for this system holds.

This paper calculates the electric and magnetic fields and the Poynting
vector around two infinitely long parallel cylindrical conductors,
carrying a DC current. Also the charges on the surface of the wire are
calculated, and their distribution is visualized. The wire is assumed to
be perfectly electrically conducting. Furthermore, the Hall effect is
ignored. In the literature [S.J. Orfanidis, Electromagnetic waves and
antennas, 2008], the problem of determining the electric field is
usually tackled using an equivalent model consisting of two line charge
densities, producing the same electric field. In this work, the Laplace
equation is rigorously solved. The authors found no work explaining the
solution of the Laplace equation with boundary conditions for this
problem and hence thought it was useful to dedicate a paper to this
topic. The method of separation of variables is employed and a bipolar
coordinate system is used. After solving the appropriate Sturm-Liouville
problems, the scalar potential is obtained. Taking the gradient yields
the electric field.
Contribution to the Topical Issue "Numelec 2012", Edited by Adel Razek.

Interest in the surface charges on circuits, and their utility in the conceptual understanding of circuit behavior, has recently increased. Papers and textbooks have discussed surface charges either with qualitative diagrams or analytic results for very special geometries. Here, I present the results of numerical calculations showing the surface charges on several simple resistor-capacitor circuits. Surface charges are seen to guide the motion of charges and create the appropriate electric potential and Poynting vectors for the circuit, and hence are an important factor in the teaching of circuit theory.

In an effort to clarify the role of surface charges on the conductors of
elementary electric circuits and the electric fields in the space around
them, we present a quantitative analysis of (two-dimensional) circular
current loops. It is also noted that, in general, lines of Poynting flux
lie in the equipotential surfaces of quasistatic systems.

The significance of the surface electric charge densities associated
with current-carrying circuits is often not appreciated. In general, the
conductors of a current-carrying circuit must have nonuniform surface
charge densities on them (1) to maintain the potential around the
circuit, (2) to provide the electric field in the space outside the
conductors, and (3) to assure the confined flow of current. The surface
charges and associated electric field can vary greatly, depending on the
location and orientation of other parts of the circuit. We illustrate
these ideas with a circuit consisting of a resistor and a battery
connected by wires and other conductors, in a geometry that permits
solution with a Fourier-Bessel series, while giving flexibility in
choice of wire and resistor sizes and location of the battery. Plots of
the Poynting vector graphically demonstrate energy flow from the battery
to the resistive elements. For a resistor with a large resistance, the
potentials and surface charge densities around the current-carrying
circuit are nearly the same as for the open circuit with the resistor
removed. For such resistors, the capacitance of a resistor and its
adjacent elements, defined in terms of the surface and interface charges
present while current flows, is roughly the same as the capacitance of
the adjacent elements of the open circuit alone. The discussion is in
terms of time-independent currents and voltages, but applies also to
low-frequency ac circuits.

By a general theorem connecting the steady-state electrical potential function for the region surrounding a system of long, parallel homogeneous conductors carrying steady currents to the solutions for the electrostatic potential in the given geometry, it is shown that the density of the surfacecharges on the conductors varies linearly with distance along the direction of their common axis.

We calculate the surface charges, potentials, and fields in a long
cylindrical coaxial cable with inner and outer conductors of finite
conductivities and finite areas carrying a constant current. It is shown
that there is an electric field outside the return conductor

- A Assis
- J A Hernandes

Assis A K T and Hernandes J A 2007 The Electric Force of a Current (Montreal: Apeiron)