# Second Gradient Electromagnetostatics: Electric Point Charge, Electrostatic and Magnetostatic Dipoles

^{*}

## Abstract

**:**

## 1. Introduction

- The field must be finite at $r=0$;
- The field must be everywhere continuous;
- The self-energy of the field must be finite.

## 2. Second Gradient Electromagnetostatics

- (1)
- ${\ell}_{1}^{4}>4{\ell}_{2}^{4}$:The length scales ${a}_{1}$ and ${a}_{2}$ are real and distinct and they read$$\begin{array}{cc}\hfill {a}_{1,2}& ={\ell}_{1}\phantom{\rule{0.166667em}{0ex}}\sqrt{\frac{1}{2}\pm \frac{1}{2}\phantom{\rule{0.166667em}{0ex}}\sqrt{1-4{\left(\frac{{\ell}_{2}}{{\ell}_{1}}\right)}^{\phantom{\rule{-0.166667em}{0ex}}4}}}\hfill \end{array}$$
- (2)
- ${\ell}_{1}^{4}=4{\ell}_{2}^{4}$:The length scales ${a}_{1}$ and ${a}_{2}$ are real and equal,$$\begin{array}{c}\hfill {a}_{1}={a}_{2}=\frac{{\ell}_{1}}{\sqrt{2}}={\ell}_{2}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$
- (3)
- ${\ell}_{1}^{4}<4{\ell}_{2}^{4}$:The two length scales ${a}_{1}$ and ${a}_{2}$ are complex conjugate,$$\begin{array}{cc}\hfill {a}_{1,2}& =A\pm \mathrm{i}B\phantom{\rule{0.166667em}{0ex}}\hfill \end{array}$$$$\begin{array}{c}\hfill A={\ell}_{2}\phantom{\rule{0.166667em}{0ex}}\sqrt{\frac{1}{2}+\frac{{\ell}_{1}^{2}}{4{\ell}_{2}^{2}}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}B={\ell}_{2}\phantom{\rule{0.166667em}{0ex}}\sqrt{\frac{1}{2}-\frac{{\ell}_{1}^{2}}{4{\ell}_{2}^{2}}}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$There is no limit to the Bopp–Podolsky theory. For generalized electrodynamics, this case leads to Green functions having a time dependence that increases exponentially, an acausal propagation and complex mass terms (e.g., [18,48,49]). The dispersion relations of the vacuum, analogous to those computed in [45], have complex coefficients, suggesting instabilities or dissipation in the vacuum.The possible negative sign in the first gradient term of the Lagrangian mentioned above also yields complex ${a}_{1}$ and ${a}_{2}$ and thus has similar consequences.

## 3. Green Functions in Second Gradient Electromagnetostatics

#### 3.1. Green Functions

#### 3.2. Derivatives of the Green Function ${G}^{L\Delta}$

## 4. Electromagnetic Fields in Second Gradient Electromagnetostatics

#### 4.1. Electric Point Charge

#### 4.2. Electric Dipole

#### 4.3. Magnetic Dipole

#### 4.4. Some Qualitative Side-Effects of Regularization

#### 4.5. Reinterpretation of Second Gradient Electromagnetostatics as Electromagnetostatics with Extended Charge and Current Densities

## 5. Electromagnetic Fields in First Gradient Electromagnetostatics (Bopp–Podolsky Electromagnetostatics)

#### 5.1. Electric Point Charge

#### 5.2. Electric Dipole

#### 5.3. Magnetic Dipole

## 6. Electromagnetic Fields in Classical Maxwell Electromagnetostatics

#### 6.1. Electric Point Charge

#### 6.2. Electric Dipole

#### 6.3. Magnetic Dipole

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Plot of the Green function ${G}^{L}$ for ${a}_{1}=2{a}_{2}$ in second gradient electromagnetostatics and of the Green function ${G}^{\mathrm{H}}$.

**Figure 2.**Plot of the auxiliary functions ${f}_{0}$, ${f}_{1}$, ${f}_{2}$ and ${f}_{3}$ for ${a}_{1}=2{a}_{2}$ in second gradient electromagnetostatics.

**Figure 3.**Plots of the radial functions in second gradient electromagnetostatics (2nd GEM) for ${a}_{1}=2{a}_{2}$, Bopp–Podolsky electromagnetostatics (BPT) and classical Maxwell electromagnetostatics (Maxwell): (

**a**) ${f}_{0}/R$, (

**b**) ${f}_{1}/{R}^{2}$, (

**c**) ${f}_{1}/{R}^{3}$, (

**d**) ${f}_{2}/{R}^{3}$, (

**e**) ${f}_{2}/{R}^{4}$ and (

**f**) ${f}_{3}/{R}^{4}$.

**Figure 4.**Electric potential $\varphi \phantom{\rule{0.166667em}{0ex}}\frac{4\pi {\epsilon}_{0}\ell}{q}$ (

**a**,

**c**) or $\varphi \phantom{\rule{0.166667em}{0ex}}\frac{4\pi {\epsilon}_{0}{a}_{1}}{q}$ (

**e**), and electric field strength $\mathit{E}\phantom{\rule{0.166667em}{0ex}}\frac{4\pi {\epsilon}_{0}{\ell}^{2}}{q}$ (

**b**,

**d**) or $\mathit{E}\phantom{\rule{0.166667em}{0ex}}\frac{4\pi {\epsilon}_{0}{a}_{1}^{2}}{q}$ (

**f**), in classical (

**a**,

**b**), Bopp–Podolsky (

**c**,

**d**) and second gradient electromagnetostatics (

**e**,

**f**) for an electric point charge and for ${x}_{2}=0$. Arrows indicate field direction, the color its absolute value. Note that in (

**a**,

**b**) the color scale fails to display the singularities.

**Figure 5.**Electric potential $\varphi \phantom{\rule{0.166667em}{0ex}}\frac{4\pi {\epsilon}_{0}{\ell}^{2}}{p}$ (

**a**,

**c**) or $\varphi \phantom{\rule{0.166667em}{0ex}}\frac{4\pi {\epsilon}_{0}{a}_{1}^{2}}{p}$ (

**e**), and electric field strength $\mathit{E}\phantom{\rule{0.166667em}{0ex}}\frac{4\pi {\epsilon}_{0}{\ell}^{3}}{p}$ (

**b**,

**d**) or $\mathit{E}\phantom{\rule{0.166667em}{0ex}}\frac{4\pi {\epsilon}_{0}{a}_{1}^{3}}{p}$ (

**f**) in classical (

**a**,

**b**), Bopp–Podolsky (

**c**,

**d**) and second gradient electromagnetostatics (

**e**,

**f**) for an electric dipole with $\mathit{p}=p{\mathit{e}}_{3}$, ${\mathit{e}}_{3}$ being the unit vector along the third coordinate, for ${x}_{2}=0$. Arrows indicate field direction, the color its absolute value. Note that in (

**a**,

**b**,

**d**) the color scale fails to display the singularities.

**Figure 6.**Magnetic vector potential $\mathit{A}\phantom{\rule{0.166667em}{0ex}}\frac{4\pi {\ell}^{2}}{{\mu}_{0}m}$ (

**a**,

**c**) or $\mathit{A}\phantom{\rule{0.166667em}{0ex}}\frac{4\pi {a}_{1}^{2}}{{\mu}_{0}m}$ (

**e**), and magnetic field strength $\mathit{B}\phantom{\rule{0.166667em}{0ex}}\frac{4\pi {\ell}^{3}}{{\mu}_{0}m}$ (

**b**,

**d**) or $\mathit{B}\phantom{\rule{0.166667em}{0ex}}\frac{4\pi {a}_{1}^{3}}{{\mu}_{0}m}$ (

**f**) in classical (

**a**,

**b**), Bopp–Podolsky (

**c**,

**d**) and second gradient electromagnetostatics (

**e**,

**f**) for a magnetic dipole with $\mathit{m}=m{\mathit{e}}_{3}$ aligned along the third coordinate direction, for ${x}_{3}=0$ (left) and ${x}_{2}=0$ (right). Arrows indicate field direction, the color its absolute value. Note that in (

**a**,

**b**,

**d**) the color scale fails to display the singularities.

**Figure 7.**Plot of the auxiliary functions ${f}_{0}$, ${f}_{1}$, ${f}_{2}$ and ${f}_{3}$ in Bopp–Podolsky electromagnetostatics.

**Table 1.**Comparison of the near-field behaviors of the electromagnetic fields of an electric point charge, an electrostatic dipole and a magnetostatic dipole.

Theory | Electric Point Charge | Electric and Magnetic Dipoles | ||
---|---|---|---|---|

$\mathbf{\varphi}$ | $\mathit{E}$ | $\mathbf{\varphi}$, $\mathit{A}$ | $\mathit{E}$, $\mathit{B}$ | |

Maxwell theory | $1/R$ | $1/{R}^{2}$ | $1/{R}^{2}$ | $1/{R}^{3}$ and $\delta \left(\mathit{R}\right)$ |

Bopp–Podolsky theory | finite | discontinuity | discontinuity | $1/R$ |

Second gradient theory | finite | approaching zero | approaching zero | finite |

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**MDPI and ACS Style**

Lazar, M.; Leck, J.
Second Gradient Electromagnetostatics: Electric Point Charge, Electrostatic and Magnetostatic Dipoles. *Symmetry* **2020**, *12*, 1104.
https://doi.org/10.3390/sym12071104

**AMA Style**

Lazar M, Leck J.
Second Gradient Electromagnetostatics: Electric Point Charge, Electrostatic and Magnetostatic Dipoles. *Symmetry*. 2020; 12(7):1104.
https://doi.org/10.3390/sym12071104

**Chicago/Turabian Style**

Lazar, Markus, and Jakob Leck.
2020. "Second Gradient Electromagnetostatics: Electric Point Charge, Electrostatic and Magnetostatic Dipoles" *Symmetry* 12, no. 7: 1104.
https://doi.org/10.3390/sym12071104