Electromagnetism: Maxwell Equations, Wave Propagation and Emission

Chapter 2

Electrostatics in Vacuum

The interaction of electric charges, as expressed by Coulomb force, is formulated according to the Newtonian concept of action-at-a-distance: if a charge q′ is produced at r′ at a time t′, a charge q located at r feels the action of q′ instantaneously, whatever the distance |r − r′| and the medium that separates the charges. The concept of field was developed by Faraday, Maxwell, Lorentz, Einstein, and many others. In modern physics, all interactions are conceived as local, i.e. involving quantities defined at the same point r and at the same time t. Fields are physical entities that are endowed with energy, momentum, etc., and they may propagate with some finite speed as waves. Furthermore, in quantum theory, the same objects (electrons for instance) have both particle and wave properties.

In this chapter, we introduce the concepts of electric field and potential, we derive the fundamental equations of electrostatics in vacuum, and we discuss some of their properties and the concept of electrostatic energy.

2.1. Electric forces and field

In a famous experiment, Coulomb used a torsion balance to measure the force of interaction of electric charges. He verified that a small charge q₁ acts on a small charge q₂ situated at a distance r with a force F_E = K_oq₁q₂/r² oriented along the line joining the charges. This force is repulsive between like charges and attractive between unlike charges. It has a similar mathematical form to Newton’s law of universal ...