8.1 Electric Charge and Electric Force

Charge is measured in coulombs (C). It comes in two kinds, positive and negative. A proton carries +e, an electron carries −e and a neutron has no charge, where e = 1.60 × 10⁻¹⁹ C is the elementary charge: the size of the charge on one proton or one electron.

Any object's net charge is a whole-number multiple of e, because it's made of a whole number of extra or missing electrons. We say charge is quantized. One coulomb is a huge amount, so real problems use microcoulombs (1 μC = 10⁻⁶ C) and nanocoulombs (1 nC = 10⁻⁹ C).

A point charge is a charged object small enough, compared with the distances in the problem, that you can treat all its charge as sitting at one point.

The force between two point charges has magnitude $$F_E = \frac{1}{4\pi\varepsilon_0}\frac{\lvert q_1 q_2 \rvert}{r^2} = k\frac{\lvert q_1 q_2 \rvert}{r^2}$$

Here r is the distance between the charges and k ≈ 9.0 × 10⁹ N·m²/C². The constant ε₀ = 8.85 × 10⁻¹² C²/(N·m²) is the permittivity of free space. Think of it as the number that sets how strongly charges interact across empty space. Both constants are on the equation sheet, and you'll use whichever one makes later formulas tidier.

The force acts along the line joining the charges. Like charges repel and opposite charges attract. By Newton's third law, the two charges feel forces of equal size in opposite directions, even if one charge is much bigger than the other.

When several charges act on one charge, each pair interacts as if the others weren't there. The net force is the vector sum of the separate Coulomb forces. This is called superposition.

The method: find the magnitude of each force with Coulomb's law (use magnitudes of the charges), decide each direction from attract or repel, break each force into x- and y-components, then add the components. The exam keeps these to four or fewer charges, unless a symmetric arrangement makes most of the work cancel.

Both forces fall off as 1/r², but they differ in two ways. Gravity only ever attracts, while the electric force can attract or repel. And the electric force is far stronger. For two protons, the electric repulsion is about 10³⁶ times their gravitational attraction. That's why you can ignore gravity for electrons, protons and ions in electric problems unless the question says otherwise.

Gravity still wins for planets and people, because large objects are almost perfectly neutral: their positive and negative charges cancel, while mass only adds up.

Electric forces also hide inside many everyday forces. The normal force, friction and tension all come from electric pushes and pulls between the atoms of surfaces in contact. There are far too many atoms to track one by one, so you treat these as contact forces instead.

In a conductor, such as a metal, some electrons are free to move through the whole material. Extra charge placed on a conductor spreads out quickly. In an insulator, such as glass, rubber or plastic, electrons are bound to their atoms, so extra charge mostly stays where you put it. This difference explains how objects get charged (8.2) and how conductors behave in fields (10.1).

Materials also differ in permittivity: how easily a field can polarize them, meaning how far it can shift their electrons to one side. Empty space has the fixed value ε₀. Any material's permittivity is different, because it depends on how its atoms and electrons are arranged and how freely those electrons can rearrange. You'll use this idea with dielectrics in capacitors (10.4).