22.4.1 Units of Charge

Coulomb had no unit of charge, so he was unable to determine a value for $K$ , whose numerical value depends on the units of both charge and distance. The SI unit of charge, the coulomb (C), is based on defining the fundamental unit of charge $e$ to be exactly $1.602 176 634 \times 10^{- 19} C$ . Stated another way, 1 C is the net charge of approximately $6.2415 \times 10^{18}$ protons. For calculations, we will use

e = 1.60 \times 10^{- 19} C

This is a very small amount of charge.

NOTE The amount of charge produced by friction is typically in the range 1 nC $(10^{- 9} C)$ to 100 nC. This is an excess or deficit of $10^{10}$ to $10^{12}$ electrons.

Once the unit of charge is established, torsion balance experiments such as Coulomb’s can be used to measure the electrostatic constant $K$ . In SI units

K = 8.99 \times 10^{9} N m^{2} / C^{2}

It is customary to round this to $K = 9.0 \times 10^{9} N m^{2} / C^{2}$ for all but extremely precise calculations, and we will do so.

Surprisingly, we will find that Coulomb’s law is not explicitly used in much of the theory of electricity. While it is the basic force law, most of our future discussion and calculations will be of things called fields and potentials. It turns out that we can make many future equations easier to use if we rewrite Coulomb’s law in a somewhat more complicated way. Let’s define a new constant, called the permittivity constant $ϵ_{0}$ (pronounced “epsilon zero” or “epsilon naught”), as

ϵ_{0} = \frac{1}{4 π K} = 8.85 \times 10^{- 12} C^{2} / N m^{2}

Rewriting Coulomb’s law in terms of $ϵ_{0}$ gives us

F = \frac{1}{4 π ϵ_{0}} \frac{| q_{1} | | q_{2} |}{r^{2}}

(22.3)

It will be easiest when using Coulomb’s law directly to use the electrostatic constant $K$ . However, in later chapters we will switch to the second version with $ϵ_{0}$ .

A photograph of a toner used in laser printers. — Coulomb’s law is at work in the operation of photocopiers and laser printers. The process depends on the fact that a few materials—notably the element selenium—are *photoconductors*, which means that they are insulators in the dark but conductors in bright light.

First, a selenium coating on a metal plate is given a uniform positive charge. Then a laser beam illuminates the plate where the image is bright. The selenium becomes conductive in these areas and loses its charge to the metal plate. Charge remains where the image is dark, forming a *latent image*. Negatively charged toner particles (microscopic beads of colored plastic resin) are attracted to and stick to the positive latent image. The image is then transferred to a positively charged sheet of paper that draws the negatively charged toner particles away. Finally, heat and pressure fuse the toner particles to the paper. In practice this is done with belts and rollers, not fixed plates, but the idea is the same.

The figure consists of four parts labeled “A” to D that illustrate the working of Coulomb’s law in a laser printer. — Coulomb’s law is at work in the operation of photocopiers and laser printers. The process depends on the fact that a few materials—notably the element selenium—are *photoconductors*, which means that they are insulators in the dark but conductors in bright light.

First, a selenium coating on a metal plate is given a uniform positive charge. Then a laser beam illuminates the plate where the image is bright. The selenium becomes conductive in these areas and loses its charge to the metal plate. Charge remains where the image is dark, forming a *latent image*. Negatively charged toner particles (microscopic beads of colored plastic resin) are attracted to and stick to the positive latent image. The image is then transferred to a positively charged sheet of paper that draws the negatively charged toner particles away. Finally, heat and pressure fuse the toner particles to the paper. In practice this is done with belts and rollers, not fixed plates, but the idea is the same.

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