Advances in High Energy Physics

Volume 2015, Article ID 180185, 7 pages

http://dx.doi.org/10.1155/2015/180185

## Analytical Calculation of Stored Electrostatic Energy per Unit Length for an Infinite Charged Line and an Infinitely Long Cylinder in the Framework of Born-Infeld Electrostatics

Department of Physics, Faculty of Sciences, Arak University, Arak 38156-8-8349, Iran

Received 28 December 2014; Accepted 7 March 2015

Academic Editor: George Siopsis

Copyright © 2015 S. K. Moayedi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

More than 80 years ago, Born-Infeld electrodynamics was proposed in order to remove the point charge singularity in Maxwell electrodynamics. In this work, after a brief introduction to Lagrangian formulation of Abelian Born-Infeld model in the presence of an external source, we obtain the explicit forms of Gauss’s law and the energy density of an electrostatic field for Born-Infeld electrostatics. The electric field and the stored electrostatic energy per unit length for an infinite charged line and an infinitely long cylinder in Born-Infeld electrostatics are calculated. Numerical estimations in this paper show that the nonlinear corrections to Maxwell electrodynamics are considerable only for strong electric fields. We present an action functional for Abelian Born-Infeld model with an auxiliary scalar field in the presence of an external source. This action functional is a generalization of the action functional which was presented by Tseytlin in his studies on low energy dynamics of *D*-branes (Nucl. Phys. B469, 51 (1996); Int. J. Mod. Phys. A 19, 3427 (2004)). Finally, we derive the symmetric energy-momentum tensor for Abelian Born-Infeld model with an auxiliary scalar field.

#### 1. Introduction

Maxwell electrodynamics is a very successful theory which describes a wide range of macroscopic phenomena in electricity and magnetism. On the other hand, in Maxwell electrodynamics, the electric field of a point charge at the position of the point charge is singular; that is, Also, in Maxwell electrodynamics, the classical self-energy of a point charge is More than 80 years ago, Born and Infeld proposed a nonlinear generalization of Maxwell electrodynamics [1]. In their generalization, the classical self-energy of a point charge was a finite value [1–6]. Recent studies in string theory show that the dynamics of electromagnetic fields on -branes can be described by Born-Infeld theory [7–10]. In a paper on Born-Infeld theory [8], the concept of a BIon was introduced by Gibbons. BIon is a finite energy solution of a nonlinear theory with a distributional source. Today, many physicists believe that the dark energy in our universe can be described by a Born-Infeld type scalar field [11]. The authors of [12] have presented a non-Abelian generalization of Born-Infeld theory. In their generalization, they have found a one-parameter family of finite energy solutions in the case of the gauge group. In 2013, Hendi [13] proposed a nonlinear generalization of Maxwell electrodynamics which is called exponential electrodynamics [14, 15]. The black hole solutions of Einstein’s gravity in the presence of exponential electrodynamics in a 3+1-dimensional spacetime are obtained in [13]. In 2014, Gaete and Helayël-Neto introduced a new generalization of Maxwell electrodynamics which is known as logarithmic electrodynamics [14]. They proved that the classical self-energy of a point charge in logarithmic electrodynamics is a finite value. Recently, a novel generalization of Born-Infeld electrodynamics is presented by Gaete and Helayël-Neto in which the authors show that the field energy of a point-like charge is finite only for Born-Infeld like electrodynamics [15]. In [16], a nonlinear model for electrodynamics with manifestly broken gauge symmetry is proposed. In the above mentioned model, for nonlinear electrodynamics, there are nonsingular solitonic solutions which describe charged particles. Another interesting theory of nonlinear electrodynamics was proposed and developed by Heisenberg and his students Euler and Kockel [17–19]. They showed that classical electrodynamics must be corrected by nonlinear terms due to the vacuum polarization effects. In [20], the charged black hole solutions for Einstein-Euler-Heisenberg theory are obtained. There are three physical motivations in writing this paper. First, the exact solutions of nonlinear field equations are very important in theoretical physics. These solutions help us to obtain a better understanding of physical reality. According to the above statements, we attempt to obtain particular cylindrically symmetric solutions in Born-Infeld electrostatics. The search for spherically symmetric solutions in Born-Infeld electrostatics will be discussed in future works. Second, we want to show that the nonlinear corrections in electrodynamics are considerable only for very strong electric fields and extremely short spatial distances. Third, we hope to remove or at least modify the infinities which appear in Maxwell electrostatics. This paper is organized as follows. In Section 2, we study Lagrangian formulation of Abelian Born-Infeld model in the presence of an external source. The explicit forms of Gauss’s law and the energy density of an electrostatic field for Born-Infeld electrostatics are obtained. In Section 3, we calculate the electric field together with the stored electrostatic energy per unit length for an infinite charged line and an infinitely long cylinder in Born-Infeld electrostatics. Summary and conclusions are presented in Section 4. Numerical estimations in Section 4 show that the nonlinear corrections to electric field of infinite charged line at large radial distances are negligible for weak electric fields. There are two appendices in this paper. In Appendix A, a generalized action functional for Abelian Born-Infeld model with an auxiliary scalar field in the presence of an external source is proposed. In Appendix B, we obtain the symmetric energy-momentum tensor for Abelian Born-Infeld model with an auxiliary scalar field. We use SI units throughout this paper. The metric of spacetime has the signature .

#### 2. Lagrangian Formulation of Abelian Born-Infeld Model with an External Source

The Lagrangian density for Abelian Born-Infeld model in a -dimensional spacetime is [1–6]where is the electromagnetic field tensor and is an external source for the Abelian field . The parameter in (3) is called the nonlinearity parameter of the model. In the limit , (3) reduces to the Lagrangian density of the Maxwell field; that is,where is the Maxwell Lagrangian density. The Euler-Lagrange equation for the Born-Infeld field isIf we substitute Lagrangian density (3) in the Euler-Lagrange equation (5), we will obtain the inhomogeneous Born-Infeld equations as follows:The electromagnetic field tensor satisfies the Bianchi identity:Equation (7) leads to the homogeneous Maxwell equations. In -dimensional spacetime, the components of can be written as follows:Using (8), (6) and (7) can be written in the vector form as follows:The symmetric energy-momentum tensor for the Abelian Born-Infeld model in (3) has been obtained by Accioly [21] as follows:where . The classical Born-Infeld equations (9) for an electrostatic field areEquations (11) and (12) are fundamental equations of Born-Infeld electrostatics [2]. Using divergence theorem, the integral form of (11) can be written in the formwhere is the three-dimensional volume enclosed by a two-dimensional surface . Equation (13) is Gauss’s law in Born-Infeld electrostatics. Using (8) and (10), the energy density of an electrostatic field in Born-Infeld theory is given byIn the limit , the modified electrostatic energy density in (14) smoothly becomes the usual electrostatic energy density in Maxwell theory; that is,

#### 3. Calculation of Stored Electrostatic Energy per Unit Length for an Infinite Charged Line and an Infinitely Long Cylinder in Born-Infeld Electrostatics

##### 3.1. Infinite Charged Line

Let us consider an infinite charged line with a uniform positive linear charge density which is located on the -axis. Now, we find an expression for the electric field at a radial distance from the -axis. Because of the cylindrical symmetry of the problem, the suitable Gaussian surface is a circular cylinder of radius and length , coaxial with the -axis (see Figure 1).