# 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

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**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.

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**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