Chapter 1: Introduction, Vector Functions and Electrostatics

Vector Functions

Vector functions are simply functions that can be expressed like this:

$$\mathbf{F}(x, y, z) = F_x(x, y, z) \hat{i} + F_y(x, y, z) \hat{j} + F_z(x, y, z) \hat{k}.$$

They specify a magnitude and direction for every point $(x, y, z)$.

Electrostatics

A fundamental property of nature is “charge”. There are two kinds of charge - positive/negative.

Coulomb’s law states that the force between two charges $q$ and $q_0$ is:

$$\mathbf{F_q} = \frac{1}{4 \pi \epsilon_0} \frac{qq_0}{r^2} \hat{u}$$

where $\hat{u}$ points from $q_0$ to $q$.

The Electric field is defined by:

$$\mathbf{E} = \frac{\mathbf{F_q}}{q}.$$

Another very important property is superposition. Superposition states that the presence of other forces does not affect the original force itself, allowing us to simply add forces vectorially.