In parallel, each resistor gets its own path between the same two points. That changes everything you just learned about series circuits: now the voltage is shared and the currents add.
Why it works
Each branch connects directly across the same two points, so each has the same voltage $V$. Each branch's current is $V/R_i$, and at the junction the currents add: $I = \dfrac{V}{R_1} + \dfrac{V}{R_2} + \cdots$. If one resistor $R_{\text{eq}}$ is to draw the same $I$, then $\dfrac{1}{R_{\text{eq}}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \cdots$. Every extra branch is an extra path, so the total current goes up and $R_{\text{eq}}$ goes down, below even the smallest resistor.
Key idea
Parallel: same voltage across each branch; the branch currents add; $\dfrac{1}{R_{\text{eq}}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \cdots$. For two: $R_{\text{eq}} = \dfrac{R_1R_2}{R_1 + R_2}$. For $n$ equal resistors: $R/n$.
Worked examples
Worked example: Two branches
A $6\,\Omega$ and a $3\,\Omega$ resistor are in parallel on a $12\,\text{V}$ battery. Find the branch currents, the total current and $R_{\text{eq}}$.
- Branch currents: $\dfrac{12}{6} = 2.0\,\text{A}$ and $\dfrac{12}{3} = 4.0\,\text{A}$.
- Total: $2.0 + 4.0 = 6.0\,\text{A}$.
- $R_{\text{eq}} = \dfrac{6 \times 3}{6 + 3} = 2.0\,\Omega$. Check: $12/2.0 = 6.0\,\text{A}$.
Answer: $2.0\,\text{A}$, $4.0\,\text{A}$; $6.0\,\text{A}$; $2.0\,\Omega$
Worked example: Three branches
Find $R_{\text{eq}}$ for $12$, $6$ and $4\,\Omega$ in parallel.
- $\dfrac{1}{R_{\text{eq}}} = \dfrac{1}{12} + \dfrac{1}{6} + \dfrac{1}{4} = \dfrac{1 + 2 + 3}{12} = \dfrac{6}{12}$.
- $R_{\text{eq}} = 2.0\,\Omega$, less than the smallest branch, $4\,\Omega$.
Answer: $2.0\,\Omega$
Common mistake
Thinking that adding a resistor in parallel raises the total resistance, because there is “more resistance” in the circuit. A new branch is a new path, so more current flows from the same battery, and that means less total resistance. Check: $R_{\text{eq}}$ of a parallel group is always smaller than its smallest branch.
A voltmeter measures the voltage between two points, so it goes in parallel across the part it measures. A good voltmeter has a very large resistance, so almost no current is diverted through it.
The outlets in a house are wired in parallel, so every appliance gets the full voltage and works whether or not the others are on.