One of the ultimate purposes of studying electricity and charge is to be able to understand electric circuits. An electrical circuit is a network of electrical components and are responsible for the functionality of all digital technology. Every circuit must have a source of power (often a battery), wires to allow the flow of charge, one or more loads (capacitors, resistors, inductors), and an output. Circuits rely on moving charge to function. This moving charge is known as current. More specifically, current is a rate of the flow of charge through a specific point in a circuit. The unit for current is the ampere, A.
I=ΔQ/Δt
Current is induced when a voltage is present. Current density, J, is a measure of current flowing through a designated cross sectional area.
J=I/A
The natural work of current is always from higher to lower potential.
Resistors are the simplest type of electrical load and are used to control the current through a circuit. Think of a resistor as a piece of sponge stuck inside of a hose. Water will still flow through the hose, but the flow will slow down at the sponge, and the pressure difference will be larger for a thicker sponge than for a thinner one. Basically, a resistor resists the flow of charge.
The potential drop at a resistor in terms of the current flowing through it and its resistance is
V=iR
This relationship between voltage, current, and resistence is known as Ohm's Law. For resistors in series, the total resistance is simply equal to the sum of the individual resistances.
Req=R1+R2+...
Req is the equivalent resistance, or the total resistance of the circuit. Resistors in parallel sum differently. For resistors in parallel, there resistance combined becomes
1/Req=1/R1+1/R2+...
Capacitors were discussed more in detail here. In brief summary though, capacitors store charge. When they are uncharged, they behave as a wire and can short a circuit. However, once they are fully charged, current can no longer run through them and they behave as a broken wire. The equations for capacitors are as follows:
C=(ΚεoA)/d if there is a dielectric
C=(εoA)/d if there is only air
Q=CV
C=Q/V
U=(QV)/2=(CV2)/2
C=C1+C2+... for capacitors in parallel
1/Ctotal=1/C1+1/C2+... for capacitors in series
Like resistors, if a circuit includes some capacitors in series and some in parallel, the equivalent capacitance can be found simply by adding the equivalent capacitances of each occurence.
An inductor is just a wire looped around a tube and is responsible for opposing changes in current.
The number of loops around the tube is called N. More about the theory on conductors will be discussed in a later section after magnetism, since inductors rely heavily on that topic. For now, know that all inductors have an inductance L, and the voltage drop at an inductor in a circuit is
V=L(di/dt)
The easiest way to understand how to analyze a circuit is to use a water flow analogy. Consider a single pipe of uniform cross section with some water flowing through it at a constant flow rate. The flow rate at one point of that pipe will be equal to the flow rate at any other point within that pipe. Now, imagine that this pipe suddenly breaks off into two pipes like a fork in the road. One of these pipes which branch off from the main one is much narrower than the other. So, the water flow which first starts in the main pipe will break off into the two branches at different amounts. The total amount of flow through the branched pipes is equal to the total flow before the water split at the node.
In this analogy, the water flow is the current, the point at which the larger pipe breaks off into two smaller pipes is the node, and the different sizes of the smaller pipes which therefore affect the flow of water through them represents an electric load of varying magnitude. In the next section, we will discuss more on how to analyze an electrical circuit and do several examples.