Series and parallel circuits

Key ideas

Components can be wired in series (one path) or parallel (branching paths). The standard focuses on parallel circuits with resistor(s) in series with the source — a common real layout.

Series rules

• Same current everywhere: $I$ is the same through each component.
• Voltages add up to the supply: $V_{\text{total}} = V_1 + V_2 + \dots$
• Resistances add:

$R_T = R_1 + R_2 + \dots$

Parallel rules

• Same voltage across each branch.
• Currents add: the supply current splits between branches, $I_{\text{total}} = I_1 + I_2 + \dots$
• Combined resistance is less than the smallest branch:

$\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \dots$

:::tip After finding $\dfrac{1}{R_T}$ for a parallel section, take the reciprocal to get $R_T$ — the most common slip. And sanity-check: the parallel combination must be smaller than the smallest branch. If you get a value bigger than any branch resistor, you have made an error. :::

Practice question

Two resistors, $4.0\ \Omega$ and $4.0\ \Omega$, are connected in parallel. Find their combined resistance.

Worked solution: $\tfrac{1}{R_T} = \tfrac{1}{4} + \tfrac{1}{4} = \tfrac{2}{4}$, so $R_T = 2.0\ \Omega$ (half of a single one — equal parallel resistors halve).