Circuit Analysis: Series, Parallel, Kirchhoff's Laws, and Potential Dividers

Real circuits contain many components connected together, and to analyze them we need rules for how current, voltage, and resistance behave in different arrangements. Two basic ways to connect components are in series and in parallel.

In a series circuit, components are joined end to end so there is only one path for the current. Because charge cannot accumulate anywhere, the same current flows through every component. The total potential difference supplied by the source is shared among the components, so the individual voltages add up to the total: $V_{\text{total}} = V_1 + V_2 + V_3 + \dots$. Since the same current passes through each resistor, the total resistance is simply the sum of the individual resistances:

$$R_{\text{series}} = R_1 + R_2 + R_3 + \dots$$

Adding resistors in series