**Electricity**

Column's law describes the electric force between any distribution and number of charged particles. It is derived empirically and falls off inversely with the square of the distance, analogous to Newton's law of gravity. The practical usefulness of this law is that it allows one to determine the effect that any distribution of charge has on any other distribution of charge; this is analogous to how Newton's law of gravity allows one to determine the effects of one mass distribution on another. One very gradually learns this by solving many problems. But in this section, we'll start off with Column's law and then proceed to do some algebra and calculus in order to derive equations which describe the electric force that systems and continuous charge distributions exert on an individual particle.

Unlike the flux of the velocity field of a fluid, the notion of electric flux is much more abstract. Like concepts such as energy, it is very abstract but nonetheless very useful in analyzing certain situations. In this section, we derive the electric flux of any electric field through any arbitrary surface. Only later one, when we discuss Guass's Law, will we learn about the usefulness of this idea.

Guass's law: for any charge distribution, whether a system of a finite number of point charges or a continuous distribution of charges, if I draw any arbitrary closed surface (which can be of any shape, size, and location), then the only contributions to the electric flux through that closed surface will be due to only the charges \(q_{enc}\) enclosed within that surface and that electric flux is given by the amount \(\frac{q_{enc}}{\epsilon_0}\). It is important to appreciate the word *any* to fully grasp the applicability of Guass's law: it applies to *any* closed surface and to *any *distribution of charge.

In this lesson, we'll see that Guass' law makes the same prediction about the electric field produced by a point charge as Column's law.

**Capacitors**

Capacitance is defined as the amount of charge stored in a capacitor per volt across the capacitor. The value of the capacitance is a measure of how rapidly a capacitor stores electric potential energy as the capacitor is getting charged up.

In this lesson, we'll prove that the capacitance of a parallel-plate capacitor does not depend on the charge or voltage of the capacitor but, rather, on only the size and geometry of the capacitor.

In this lesson, we'll determine the electric field generated by a charged plate. We'll show that a charged plate generates a constant electric field. Then, we'll find the electric field produced by two, parallel, charged plates (a parallel-plate capacitor). We'll show that the electric field in between the plates has a constant magnitude \(\frac{σ}{ε_0}\). We'll also show that the direction of the electric field is a constant pointing from the positively charged plate to the negatively charged plate in a direction that is perpendicular to both plates. Lastly, we'll also prove that the electric field "outside" of the capacitor is zero everywhere.

In this lesson, we'll determine the electric potential difference (also called voltage) across any arbitrary capacitor.