Physics with Calculus/Electromagnetism/Current and Circuits

Current
Imagine a wire with charge flowing through it. The current is the rate at which charge flows through the wire. Say that there are n movable electrons per unit volume, that the electrons have a charge e, and move at a velocity in the direction of the wire with velocity v. Also assume that the wire has cross-sectional area A.  Then the charge per second passing through the surface is envA. You can easily see this if you imagine a cylinder of the wire of length v dt. It has a volume of Av dt, and by the definition of n, nAv dt electrons, and by definition of e, enAv dt charge. All of that (and only all of that) will go through the wire in the time dt. Therefore, the current, i = envA.

Notice that in our formula, even though electrons have negative charge, we regard the current as positive. This is called conventional current. It is equivalent to an equal and opposite flow of positive charge (mathematically, that means multiplying both sides of i = envA by -1).

Circuits
A circuit component is a device that sets up a relationship between voltage and current. A circuit component may have any number of terminals, and have any relationship. However, in practice, there are only a few basic circuit components. In analyzing circuits, it is highly advantageous to use only circuit components that, given the voltage at a single instant, or infinitesimal neighborhood, you can determine the current, and vice versa. Two very simple circuit elements are voltage sources and current sources. A voltage source keeps the voltage fixed and lets the current be whatever it wants. A current source keeps the current constant and lets the voltage be whatever it wants. A third simple circuit element is the resistor, which keeps the voltage proportional to the current, with the constant of proportionality called the resistance. We can write these relationships as:

V = V_0; I = I_0; V = IR

for a voltage source, current source, and resistor respectively.

A special case of the voltage source is when it keeps the voltage zero; this is called a short circuit or wire. When a current source keeps the current zero, it is called an open circuit. We may regard an open circuit as not being part of the circuit at all, and a short circuit as just combining the two circuit elements it joins.

A circuit or network is a collection of circuit components hooked together. Naturally, there will be loops and nodes, which are when two or more circuit components join together. At every node, we assume that charge does not build up, and that charge is conserved. That is, the current going in is the current going out. In any loop, we assume that an electron going around cannot speed up faster and faster ad infinitum, in other words, the voltage in a closed loop must be zero; that is, energy is conserved. These two rules, along with the voltage-current (v-i) relationships, completely describes a circuit. With these, we can find every voltage and every current.