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Electric and magnetic fields of two infinitely long parallel cylindrical conductors carrying a DC current
Abstract
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.
... A similar solution for this problem has been found recently by Engelen et al [10]. They solve essentially the same problem using bi-cylinder coordinates (which they call bipolar coordinates). ...
... The similar solution presented by Engelen et al [10] lack the dependency on z for the potential, electric field and surface charges, rendering essentially an electrostatics problem. They were probably unaware of Russel's theorem [11]. ...
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.
... mm, much smaller compared to the cylinder diameter of 9.6 mm), this effect on the field on the axis of symmetry is small. This has been verified by comparing the electric field halfway between two metal cylinders (without dielectrics) calculated for the same cylinders diameter and effective gap, (a) from the superposition principle and (b) from exact solution of the Laplace equation [23]. The difference between the two results is approximately 2%. ...
Electric field in nanosecond pulse discharges in ambient air is measured by picosecond four-wave mixing, with absolute calibration by a known electrostatic field. The measurements are done in two geometries, (a) the discharge between two parallel cylinder electrodes placed inside quartz tubes, and (b) the discharge between a razor edge electrode and distilled water surface. In the first case, breakdown field exceeds DC breakdown threshold by approximately a factor of four, 140 ±10 kV cm⁻¹. In the second case, electric field is measured for both positive and negative pulse polarities, with pulse durations of ∼10 ns and ∼100 ns, respectively. In the short duration, positive polarity pulse, breakdown occurs at 85 kV cm⁻¹, after which the electric field decreases over several ns due to charge separation in the plasma, with no field reversal detected when the applied voltage is reduced. In a long duration, negative polarity pulse, breakdown occurs at a lower electric field, 30 kV cm⁻¹, after which the field decays over several tens of ns and reverses direction when the applied voltage is reduced at the end of the pulse. For both pulse polarities, electric field after the pulse decays on a microsecond time scale, due to residual surface charge neutralization by transport of opposite polarity charges from the plasma. Measurements 1 mm away from the discharge center plane, ∼100 μm from the water surface, show that during the voltage rise, horizontal field component (Ex ) lags in time behind the vertical component (Ey ). After breakdown, Ey is reduced to near zero and reverses direction. Further away from the water surface (≈0.9 mm), Ex is much higher compared to Ey during the entire voltage pulse. The results provide insight into air plasma kinetics and charge transport processes near plasma-liquid interface, over a wide range of time scales.
... Along similar lines, in case 6 of Fig. 3·9, voltage profile between phases A and B, which are maintained at zero potential, is affected by the presence of phase C electrodes of previous and current spatial period. Analytical models have been derived for the electric potential between two cylindrical conductors, like (Engelen et al., 2013) or two-wire transmission line (Orfanidis, 2014), that can be potentially used to provide better approximations for the voltage profile closer to what FEA method provides, rather than a linear one. ...
Particulate contamination of the optical surfaces of solar collectors, often called "soiling", can have a significant deteriorating impact on energy yield due to the absorption and scattering of incident light. Soiling has more destructive effect on concentrated solar systems than on flat-plate photovoltaic panels, as the former are incapable of converting scattered sunlight. The first part of this thesis deals with the soiling losses of flat-plate photovoltaic (PV), concentrated solar power (CSP), and concentrated photovoltaic (CPV) systems in operation in several regions of the world. Influential parameters in dust accumulation losses, as well as different cleaning mechanisms in pursuit of restoring the efficiency of soiled systems, have been thoroughly investigated.
In lieu of the most commonly-practiced manual cleaning method of using high-pressure water jets, the concept of automatic dust removal using the electrostatic forces of electrodynamic screen (EDS) technology is in a developmental stage and on its way toward commercialization. This thesis provides comprehensive analytical solutions for the electric potential and electric field distribution in EDS devices having different configurations. Numerical simulations developed using finite element analysis (FEA) software have corroborated the analytical solutions which can easily be embedded into software programs for particle trajectory simulations while also providing flexibility and generality in the study on the effect of different parameters of the EDS on the electric field and ensuing dust-removal performance.
Evaluation and comparison of different repelling and attracting forces exerted on dust particles is of utmost importance to a detailed analysis of EDS performance in dust removal. Hence, the balance of electrostatic and adhesion forces, including van der Waals and capillary forces, have received significant attention in this dissertation. Furthermore, different numerical analyses have been conducted to investigate the potential causes of observed failures of EDS prototypes that functioned well in a laboratory environment but failed after outdoor exposure.
Experimental studies form the last two chapters of this dissertation. Different tests have been conducted on an EDS sample integrated with a PV cell to restore the efficiency of the cell after dust deposition. In order to evaluate the performance of the EDS in dust-particle removal, we have studied the particle size distribution on the EDS surface after each dust deposition and EDS cleaning cycle using a custom-built dust-deposition analyzer. Furthermore, we have pursued several experiments to examine how the geometric and operational EDS parameters affect particle charge via charge-to-mass-ratio measurements.
... On similar lines, in Fig. 5 case 3, when the first and second electrode are at 1 kV electric potential, the voltage profile provided by the FEA method shows a decrease in electric potential between the aforementioned electrodes down to 957.2 V while the voltage is assumed to be still at 1 kV along the x-axis in ANA method. Existing analytical models discussed in [32] and [33] have the potential to provide better approximations of the voltage profile than a linear onedcloser to what the FEA method yields. . Providing a detailed model of the voltage profile will improve the accuracy the analytical method, which can be a promising research topic for further studies. ...
The enormous potential of solar energy harvesting plants to provide clean energy is severely limited by dust accumulation on their optical surfaces. In lieu of the most commonly-practiced manual cleaning method of using high-pressure water jets, electrodynamic screen (EDS) technology offers an attractive solution for removing dust particles from optical surfaces using electrostatic forces. In this paper, the impacts of different EDS design parameters in the electric field distribution on an EDS have been studied. Furthermore, based on electric field expressions, closed-form solutions for multipolar dielectrophoretic (DEP) forces in the EDS application are provided. Detailed evaluation of the EDS performance necessitates investigation of different forces involved in the dust removal process. Different comparisons are made between repelling and attracting forces exerted on dust particles deposited on an EDS surface. These comparisons elucidate EDS performance in the removal of a given size range of dust particles. The significant detrimental impact of relative humidity upon the dust removal process is quantitatively addressed. It is shown how just a 10 percent increase in relative humidity can make the repelling force ineffective in the dust removal process.
... Along similar lines, in case 6 of Fig. 5, the voltage profile between phases A and B, which are maintained at zero potential, is affected by the presence of phase C electrodes of the previous and current spatial period. Analytical models have been derived for the electric potential between two cylindrical conductors, like [16] or two-wire transmission line [17], that can be potentially used to provide better approximations for the voltage profile closer to what FEA method provides, rather than a linear one. Detailed analytical model of the voltage profile is out of the scope of this paper and is left for further studies. ...
The electrodynamic screen, or EDS, has shown promising results in mitigation of dust accumulation losses in solar energy harvesting systems. In this paper, the electric field distributions in two EDS configurations have been thoroughly investigated. The analytical solutions for the electric potential and electric field distribution in the first EDS configuration have been provided and corroborated numerically using finite element analysis (FEA) software. The electrostatic model of second EDS configuration has been developed in the FEA software and its electric field distribution has been analyzed numerically. A comparison has been made between the two configurations regarding dust removal efficiency.
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.