Properties of Electric Charges and Coulomb's Law

$q = Ne$

Where $q$ is the total charge, $N$ is an integer, and $e$ is the elementary charge ($1.602 \times 10^{-19}$ C).

Exam Tip: Remember that while charge is quantized, the Coulomb (C) is a very large unit. In many laboratory experiments, you will deal with much smaller prefixes like microcoulombs ($\mu\text{C} = 10^{-6}\text{ C}$) or nanocoulombs ($\text{nC} = 10^{-9}\text{ C}$).

Conductors, Insulators, and Semiconductors

How charge moves through a material depends on its atomic structure. Materials are generally classified into three categories:

1. Electrical Conductors: Materials like copper, aluminum, and silver contain "free electrons" that are not bound to specific atoms and can move relatively freely. If you charge a small region of a conductor, the charge will distribute itself over the entire surface.
2. Electrical Insulators: Materials such as glass, rubber, and dry wood have electrons that are tightly bound to their atoms. When an insulator is charged by rubbing, the charge remains localized in the area that was rubbed.
3. Semiconductors: Materials like silicon and germanium fall between these two extremes, possessing electrical properties that are intermediate between insulators and conductors.

Coulomb's Law: Magnitude and Direction

To quantify the force between charges, we use Coulomb's Law. This law applies to point charges—charged particles modeled as having zero size. While large objects aren't point charges, we can often model them as a collection of many tiny point charges.

The Scalar Form (Magnitude)

The magnitude of the electric force between two stationary point charges, $q_1$ and $q_2$, separated by a distance $r$, is given by:

$F = k \frac{|q_1 q_2|}{r^2}$

In this equation, $k$ is the Coulomb constant. The constant $k$ is related to the permittivity of free space ($\epsilon_0$), which represents the ability of a vacuum to permit electric field lines:

$k = \frac{1}{4\pi\epsilon_0} \approx 8.9876 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$

The force is inversely proportional to the square of the distance ($r^2$), meaning if you double the distance, the force becomes four times weaker.

The Vector Form (Direction)

Because force is a vector, we must account for direction. To do this, we use a unit vector ($\hat{r}$), which has a magnitude of 1 and points along the line connecting the two charges. The vector form of Coulomb's Law is:

$\vec{F}_{12} = k \frac{q_1 q_2}{r^2} \hat{r}$

In this form, including the signs of the charges allows the math to naturally indicate whether the force is attractive or repulsive. If you are calculating the resultant force on a single charge being acted upon by multiple other charges, you must use vector addition.

![An illustration of two point charges, q1 and q2, showing the distance r between them and the direction of the electric force vectors acting on each charge using unit vectors.](sf-read-illust: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