# Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**sl**and

**cl**, respectively). It was no wonder that elliptic functions, with double periodicity, appeared in the solution of double lattice series, but of special interest in our case was the duality, a certain kind of similarity between lemniscate functions and their trigonometric counterparts. For definitions and properties of lemniscate functions, see, e.g., [7,8,9] or online sources such as [10]. Here we specify two fast-converging series useful for the numeric evaluation of

**sl**and

**cl**[11]:

**sl**and

**cl**expressed as Jacobian elliptic functions (see e.g., [7,10]). As an example, with $z=L/2$, for which we know that $\mathbf{sl}\left(z\right)=1$, computation of the right-hand side of Equation (1a) with $s=5$ returns the result 0.99999999378… We also calculated the first five nonzero Taylor expansion coefficients of the

**sl**function (found in the Online Encyclopedia of Integer Sequences [12] as sequence A104203) using Equation (1a) with s = 4 as: {1.000000172; −11.99999964; 3024.000001; −4,390,848; 21,224,560,896}. A corresponding result was obtained for the

**cl**function. Other series expansions for the

**cl**function may be found in [13], and these may also be used for

**sl**via the property $sl\left(z\right)=cl\left(z-L/2\right),$ where z is any complex number.

## 2. Investigation Method

## 3. Results

#### 3.1. Considered Cases

- Case A: “+” at $\left(ma,0\right)$; co-ordinates [$m$ vary, $n$ fixed at 0, only “+”]
- Case B: “+”/”−“ at even/odd $\left(ma,0\right)$; co-ordinates [$m$ vary, $n$ fixed at 0, “+”/”−“ if $m$ even/odd]
- Case C: “+”/”−“ at even/odd $\left(ma,na\right)$; co-ordinates [$m$ and $n$ vary, “+”/”−“ if $m+n$ even/odd]
- Case D: “+”/”−“ at even/odd $\left(ma,na\right)$; co-ordinates [$m$ and $n$ vary, “+”/”−“ if $m$ even/odd]

#### 3.2. Electric-Field Components as Sum Formulas

#### 3.3. Electric Potentials as Sum Formulas

#### 3.4. Closed-Form Expressions

## 4. Discussion

#### 4.1. Comparisons to Earlier Works

#### 4.2. Duality Aspect and Corollary Findings Related to Gauss’s Constant

**sl**for cos/

**cl**, with the resulting identity being reminiscent of but not equivalent to Equation (12) in [13]. We found further identities of a similar kind, involving other trigonometric functions including hyperbolic functions, and uploaded these on [11]. Similar closed-form solutions can be found by applying the superposition principle to solutions for the potentials for Cases B and C.

#### 4.3. Potential Applications and Extension to Asymmetric Planar Arrangements

## 5. Summary and Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Borwein, J.M.; Glasser, M.L.; McPhedran, R.C.; Wan, J.G.; Zucker, I.J. Lattice sums then and now. In Encyclopedia of Mathematics and Its Applications; Cambridge University Press: Cambridge, UK, 2013; ISBN 978-1-107-03990-2. [Google Scholar]
- McDonald, K.T. Notes on Electrostatic Wire Grids (Published 2003). International Note. Available online: http://www.hep.princeton.edu/~mcdonald/examples/grids.pdf (accessed on 1 June 2020).
- Weertman, J. Dislocation Based Fracture Mechanics; World Scientific Publishing Company: Singapore, 1996; ISBN 978-9-813-10497-6. [Google Scholar]
- Gradshteyn, I.S.; Ryzhik, I.M. Table of Integrals, Series and Products, 7th ed.; Academic Press, Elsevier Inc.: Cambridge, MA, USA, 2007; ISBN 978-0-12-373637-6. [Google Scholar]
- Hansen, E.R.A. A Table of Series and Products; Prentice-Hall: Upper Saddle River, NJ, USA, 1975; ISBN 978-0-13-881935-5. [Google Scholar]
- Prudnikov, A.P.; Brychkov, Y.A.; Marichev, O.I. Integrals and Series; Gordon and Breach Science Publishers: New York, NY, USA, 1986–1992; Volumes 1–3. [Google Scholar]
- Markushevich, A.I. The Remarkable Sine Functions; American Elsevier Publishing Company: Amsterdam, The Netherlands, 1966. [Google Scholar]
- Eberhard, Z. Oxford Users’ Guide to Mathematics; Oxford University Press: Oxford, UK, 2004; ISBN 978-0-198-50763-5. [Google Scholar]
- Young, R.M. Excursions in Calculus: An Interplay of the Continuous and the Discrete; Cambridge University Press: Cambridge, UK, 1992; Volume 13, ISBN 978-0-883-85317-7. [Google Scholar]
- Lemniscate Functions. E.D. Solomontsev (Originator), Encyclopedia of Mathematics. Available online: http://encyclopediaofmath.org/index.php?title=Lemniscate_functions&oldid=17903 (accessed on 27 May 2020).
- A Collection of Infinite Products and Series (Andreas Dieckmann). Available online: http://www-elsa.physik.uni-bonn.de/~dieckman/InfProd/InfProd.html (accessed on 16 May 2020).
- The Online Encyclopedia of Integer Sequences (N. J. A. Sloane). Available online: https://oeis.org (accessed on 16 May 2020).
- Boyd, J.P. New series for the cosine lemniscate function and the polynomialization of the lemniscate integral. J. Comput. Appl. Math.
**2011**, 235, 1941–1955. [Google Scholar] [CrossRef] [Green Version] - Wolfram Research, Inc. Mathematica, Version 12.1; Wolfram Research, Inc.: Champaign, IL, USA, 2020.
- Purcell, E.M. Electricity and Magnetism (Berkeley Physics Course, Vol.2); McGraw-Hill: New York, NY, USA, 1966; ISBN 978-0-070-04908-6. [Google Scholar]
- Table of Integrals (Andreas Dieckmann). Available online: http://www-elsa.physik.uni-bonn.de/~dieckman (accessed on 29 May 2020).
- Lu, C.; McDonald, K. The Electric Potential of Particle Detectors with Rectangular Cross-Section (Published 2002). Available online: http://puhep1.princeton.edu/~mcdonald/examples/iarocci.pdf (accessed on 1 June 2020).
- Orjubin, G. Closed-form expressions of the electrostatic potential close to a grid placed between two plates. J. Electrost.
**2017**, 87, 195–202. [Google Scholar] [CrossRef] - Tomitani, T. Analysis of potential distribution in a gaseous counter of rectangular cross-section. Nucl. Instr. Methods
**1972**, 100, 179–191. [Google Scholar] [CrossRef] - Cooperman, P. A theory for space-charge-limited currents with application to electrical precipitation. AIEE Trans.
**1960**, 79, 47–50. [Google Scholar] [CrossRef]

**Figure 1.**Sections of size $4\times 4$ of electric potentials corresponding to Cases (

**A**) (upper left), (

**B**) (upper right), (

**C**) (lower left) and (

**D**) (lower right). Reference points set such that $U\left(1/2,1/2\right)=0,$ and $a,$ $\lambda $ were set to 1.

**Figure 2.**Equipotential lines and electric-field vectors of potential section shown in Figure 1 for Cases (

**A**), (

**B**), (

**C**) and (

**D**) as indicated. Blue/yellow areas are below/above the chosen reference point of each potential. Space between charges shown in ochre carried potential values near zero. All potentials, lines, and vectors drawn using closed expressions from Equations (8)–(10), setting a and $\lambda $ to 1. ContourPlots with overlayed VectorPlots are shown using automatic vector scaling.

**Figure 3.**(

**left**) Field strengths and (

**right**) electric potentials of (blue line) square tube and (orange line) Geiger counter, shown decreasing from the wire edge to the grounded wall.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vigren, E.; Dieckmann, A.
Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges. *Symmetry* **2020**, *12*, 1040.
https://doi.org/10.3390/sym12061040

**AMA Style**

Vigren E, Dieckmann A.
Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges. *Symmetry*. 2020; 12(6):1040.
https://doi.org/10.3390/sym12061040

**Chicago/Turabian Style**

Vigren, Erik, and Andreas Dieckmann.
2020. "Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges" *Symmetry* 12, no. 6: 1040.
https://doi.org/10.3390/sym12061040