### Electromagnetic Forces

Electric charge is, like mass, a fundamental property—a property that isn’t explained in terms of other properties. Yes, you can explain the charge of an object in terms of the number of protons and electrons it contains (as we’ll see in chapter 4); but the fact that electrons and protons have charge isn’t explained in terms of anything else, at least in classical physics. Some fundamental particles just have charge, and others don’t, end of story.

Most macroscopic objects have no total charge, because they are comprised of atoms that contain an equal number of positively charged protons and negatively charged electrons. But electrons tend to accumulate near the surfaces of some materials, and to move away from the surfaces of other materials. So rubbing different materials together can sometimes transfer electrons from one material to another. For example, rubbing a plastic comb through your hair can transfer electrons from your hair to the comb, resulting in a negatively-charged comb and positively-charged hair.

Recall that any two objects with mass exert gravitational forces on each other. Similarly, any two charged objects exert forces on each other, and these forces are described by a law similar in form to Newton’s law of universal gravitation. The corresponding law for electric charges is named after French physicist Charles Augustin de Coulomb. Coulomb’s law says that any two charged objects exert a force on each other. The force is attractive if the objects have opposite charge (one positive and the other negative), and repulsive if they have the same charge (both positive or both negative). The strength of this force depends on their charges and the distance between them as follows:

F =
 k × c1 × c2 d2

In the above equation, F is the magnitude (strength) of the force, c1 and c2 are the charges of the objects (in coulombs), and d is the distance between them. k is Coulomb’s constant, which has the following value:

k  =  9.00 × 109 N m2/C2

Like the gravitational constant (G), Coulomb’s constant is the same at all times and places throughout the universe: it is a physical constant.

Notice the close resemblance between Coulomb’s law and Newton’s law of universal gravitation:

Newton’s law
of universal gravitation:
force =
 G × mass1 × mass2 distance2
Coulomb’s law: force =
 k × charge1 × charge2 distance2

Mass and charge are both fundamental properties, and the fundamental forces associated with them are very similar. There are several important differences, however:

1. Unlike mass, charge comes in two types: positive and negative. Objects with the same type of charge exert a repulsive force on each other, while objects with opposite charge exert an attractive force on each other. The gravitational force, in contrast, is always attractive.
2. Gravitational and electromagnetic forces differ dramatically in strength: the force of gravity is extremely weak compared to the electromagnetic force.

You can get some idea of the relative strengths of these forces by comparing the fundamental constants: G is a very tiny number, whereas k is a very large number. However, the numerical values of these constants depend on our choice of units, which is arbitrary. It is possible to choose mass units much larger than kilograms, and charge units much smaller than Coulombs, so that G and k have the same numerical value.

For a more objective comparison of the strengths of the fundamental forces, we can compare the forces exerted between fundamental particles like electrons or protons, which have both mass and charge. For example, any two protons in the universe are exerting both a gravitational force and an electromagnetic force on each other all the time. Regardless of how far apart the protons are, the strength of the electromagnetic force between them is approximately 1036 times stronger than the gravitational force between them. (That’s 1,000,000,000,000,000,000,000,000,000,000,000,000 times stronger!) With electrons, the difference between the gravitational and electromagnetic forces is even more extreme, since electrons have just as much charge as protons but far less mass.

3. Electric charges are associated with two types of fields (electric fields and magnetic fields), whereas mass is associated only with the gravitational field. We’ll learn what fields are on the next page.
4. When charged objects move, the electromagnetic force does funky things. We’ll learn about that on the next page, too.