Chapter 1 Charge

A boy appears to hover in midair, but closer inspection reveals fine silk threads suspending him above the ground. A turning wheel touches the soles of the boy’s bare feet and his hand reaches out to a little girl standing on a stool. Her dress puffs out and her hands begin to attract bits of finely chopped straw. A pretty young lady stands on the stool, and invites a suitor to kiss her in the dark. Much to his chagrin, the gentleman is thwarted by a bright spark of fire to his lips. The woman retains her virtue, and the man must reclaim his masculinity by igniting a vat of spirits with a metal sword.[1]

In the eighteenth century, such crowd-pleasing demonstrations not only showed that human beings could hold quite a bit of what’s now known as electric charge, but also that electrical effects could travel—from wheel, to boy, to girl, to woman, to man, to sword, to spirits. Electric charge is the subject of this chapter.

The “flying boy” demonstration is only one example of Stephen Gray’s extensive experimentation with threads, wires, moist fibers and electrical fire. Gray’s work established that electrical effects could travel great distances—800 feet in once case—and cast doubt on the earlier notion that electricity was firmly attached to specific materials. During the eighteenth century, investigators showed that, like heat or liquid, electricity could flow from one object to another, sometimes without direct contact. Electricity, it seemed, was like a fluid.

Charles Francois de Cisternay du Fay was among those who continued where Gray’s work left off. He noted that a glass tube could charge and then repel a light gold leaf, while resins such as amber could attract the same leaf. Such experiments led him to propose a model of electricity that consisted of two fluids: vitreous (relating to the Latin word for glass) and resinous (for resin). Du Fay further noted the difference between conductors, which he called (somewhat counterintuitively) non-electrics and insulators, which he called electrics. Du Fay reported that bodies electrified (we would now say charged) with vitreous electricity attract bodies electrified with resinous electricity but repel other bodies electrified with vitreous electricity. Du Fay hadn’t helped at all to answer the question of how electricity could act through the air. Instead, he raised another question: where did the two apparent types of electricity come from?

Benjamin Franklin built on this work and showed that while objects contain charges, they are usually neutral. His use of the less-colorful terms positive and negative for the two types of electric charges replaced du Fay’s vitreous and resinous electric fluids and conveyed the idea that a neutral body contains equal numbers of positive and negative charges. In Franklin’s view, a body becomes electrically charged because of a transfer of electricity. A negatively charged body, for example, can be thought to have an excess of negative charge or a deficiency of positive charge.

Coulomb. subsequently proposed a perfectly good rival theory to Franklin’s one-fluid theory, which was the two-fluid theory of electricity. In Franklin’s model there is one common electrical fluid, and the two types of charge merely reflect a deficit or surplus of this common fluid. On the other hand, Coulomb contended that there are separate positive and negative electric fluids, and that each fluid cancels the other when present in equal amounts. Both models have some merit, along with deficiencies, but their best features are now contained within our understanding of the atomicity of charge.[2]

In this chapter, we will:

  • Develop the ideas of charge conservation, electric current, and current density,
  • Discuss the continuity equation for electric charge, and its solutions for several simple problems.

Along the way, we also present some of the simpler solutions to the partial differential equations that govern electric charges and their flow.

[1] P. Bertucci, “Sparks in the dark: the attraction of electricity in the eighteenth century,” Endeavour, 31:3 (2007), 88-93.

[2] Roller and Roller, The Development of the Concept of Electric Charge: Electricity from the Greeks to Coulomb (Cambridge, Mass.: Harvard University Press, 1954), 80-1.