# Deriving an Electric Wave Equation from Weber’s Electrodynamics

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## Abstract

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## 1. Introduction

## 2. Weber’s Electrodynamics

## 3. Vacuum Polarization

## 4. Derivation of Electric Wave Equation

## 5. Longitudinal Electric Wave

## 6. Discussion

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Sketch of vacuum postulated as a polarizable material. The positive–negative charge pairs are not dipoles since the charges can fully overlap each other when ${\stackrel{\u20d1}{D}}_{+}=-{\stackrel{\u20d1}{D}}_{-}=0$.

**Figure 2.**Sketch of physical quantities at the origin, $\stackrel{\u20d1}{0}$, and at $\stackrel{\u20d1}{r}$.

**Figure 3.**Sketch of charges in the volume element $d\stackrel{\xb4}{V}$ on the spherical shell, and charges in the volume element $dV$ at the origin. $ds$ is an area on the shell and $dr$ is the thickness of the shell.

Symbol | Meaning | Symbol | Meaning |
---|---|---|---|

$\stackrel{\u20d1}{F}$ | Total force. | $Q$, $q$ | Electrical charges. |

${\stackrel{\u20d1}{F}}_{c}$ | Integrated static (Coulomb) force. | ${\epsilon}_{0}$ | Dielectric constant. |

${\stackrel{\u20d1}{F}}_{v}$ | Integrated velocity related force. | $c$ | Speed of light. |

${\stackrel{\u20d1}{F}}_{a}$ | Integrated acceleration related force. | $\widehat{r}$ | Unit vector. |

$\stackrel{\u20d1}{f}$ | Force between two parcels of charges. | $r$ | Distance between the two charges. |

${\stackrel{\u20d1}{f}}_{c}$ | Static (Coulomb) force. | $\stackrel{\u20d1}{v}$ | Relative velocity. |

${\stackrel{\u20d1}{f}}_{v}$ | Velocity related force. | $\stackrel{\u20d1}{a}$ | Relative acceleration. |

${\stackrel{\u20d1}{f}}_{a}$ | Acceleration related force. | $\stackrel{\u20d1}{E}$ | External electric field. |

${\stackrel{\u20d1}{D}}_{+}$,${\stackrel{\u20d1}{D}}_{-}$ | Displacement of positive and negative charges. | $g\left(\right)$ | Relationship between the electric field and the displacement field. |

${\stackrel{\u20d1}{v}}_{+}$,${\stackrel{\u20d1}{v}}_{-}$ | Velocity of positive and negative charges. | $\nabla \xb7$ | Divergence operator. |

${\stackrel{\u20d1}{a}}_{+}$,${\stackrel{\u20d1}{a}}_{-}$ | Acceleration of positive and negative charges. | $O\left({\stackrel{\u20d1}{r}}^{2}\right)$ | Higher-order terms. |

${\rho}_{+}$,${\rho}_{-}$ | Absolute density of positive and negative charges. | $\xb7$ | Dot product of vector. |

$\rho $ | Average density of positive and negative charges. | $\nabla $ | Gradient operator. |

${\rho}_{-}\left(\stackrel{\u20d1}{r}\right)$ | Density of negative charge at location $\stackrel{\u20d1}{r}$. Similar symbol for positive charge. | $dV$ | Small parcel at origin. |

${\stackrel{\u20d1}{v}}_{-}\left(\stackrel{\u20d1}{r}\right)$ | Velocity of negative charge at location $\stackrel{\u20d1}{r}$. | $d\stackrel{\xb4}{V}$ | Small parcel at location $\stackrel{\u20d1}{r}$ on the shell. |

${\stackrel{\u20d1}{a}}_{-}\left(\stackrel{\u20d1}{r}\right)$ | Acceleration of negative charge at location $\stackrel{\u20d1}{r}$. | $ds$ | Small area on the shell. |

${\rho}_{-}\left(\stackrel{\u20d1}{0}\right)$ | Density of negative charge at origin. | $dr$ | Thickness of the shell. |

${\stackrel{\u20d1}{v}}_{-}\left(\stackrel{\u20d1}{0}\right)$ | Velocity of negative charge at origin. | $\frac{\partial}{\partial t}$ | Eulerian derivative. |

${\stackrel{\u20d1}{a}}_{-}\left(\stackrel{\u20d1}{0}\right)$ | Acceleration of negative charge at origin. | $\frac{d}{dt}$ | Lagrangian derivative. |

$\chi $ | Constant scalar. | $\nabla \times $ | Curl operator. |

$\mathsf{\Phi}$ | Scalar potential. | ${\nabla}^{2}$ | Laplacian operator. |

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**MDPI and ACS Style**

Li, Q.; Maher, S.
Deriving an Electric Wave Equation from Weber’s Electrodynamics. *Foundations* **2023**, *3*, 323-334.
https://doi.org/10.3390/foundations3020024

**AMA Style**

Li Q, Maher S.
Deriving an Electric Wave Equation from Weber’s Electrodynamics. *Foundations*. 2023; 3(2):323-334.
https://doi.org/10.3390/foundations3020024

**Chicago/Turabian Style**

Li, Qingsong, and Simon Maher.
2023. "Deriving an Electric Wave Equation from Weber’s Electrodynamics" *Foundations* 3, no. 2: 323-334.
https://doi.org/10.3390/foundations3020024