Combined Resistance

Resistors slow the flow of electric current, like narrow sections of a pipe slow water. When resistors are connected together, we can replace them with one value called the combined resistance. This makes circuits easier to analyse.

Key idea: Resistance and current

Resistance is measured in ohms (Ω). For a given voltage, higher resistance means smaller current. The relationship is given by Ohm’s law: $V = IR$.

Resistors in series (one path, end-to-end)

The same current flows through each resistor. The total voltage is shared. The combined resistance is the sum:

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

Analogy: series resistors are like making a pipe longer. A longer pipe is harder to push water through, so resistance increases.

Resistors in parallel (branches side-by-side)

Each branch has the same voltage across it. The current splits and then adds back together at junctions. The combined resistance is found using reciprocals:

$$\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots$$

For two resistors: $R_{\text{total}} = \dfrac{R_1 R_2}{R_1 + R_2}$.

Analogy: parallel resistors are like adding extra lanes to a road. More lanes make it easier for cars to pass, so the total resistance is smaller than any single lane.

Common misconceptions

• Do not add resistances for parallel; use reciprocals (or the two-resistor formula).
• In parallel, the source current is the sum of branch currents; it is larger than the current in each branch.
• Voltage across parallel branches is the same; it is not shared like in series.
• In parallel, the smallest resistor takes the largest current, so it strongly reduces $R_{\text{total}}$.