Resistors in Parallel

Two resistors are in parallel when they are connected between the same two nodes: both ends of $R_1$ touch the same points as the ends of $R_2$. The current arriving at the node splits between the two branches and recombines afterwards. The key property is thus complementary to that of series connections: each resistor has the same potential difference $\Delta V$ across it.

Two resistors in parallel between nodes $A$ and $B$: same voltage $\Delta V$, the current splits between the branches.

Since the total current is the sum of the branch currents and each branch obeys Ohm’s law with the same $\Delta V$: $i = i_1 + i_2 + \cdots = \frac{\Delta V}{R_1} + \frac{\Delta V}{R_2} + \cdots$

Dividing by the common $\Delta V$ gives the reciprocal rule.

Principle — Resistors in parallel

Same voltage across them; currents add up.

Key formula

$\ev{\frac{1}{R_\text{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots}$

Adding a branch in parallel offers the current one more path: the equivalent resistance decreases. In fact $R_\text{eq}$ is always smaller than the smallest of the parallel resistances.

Summary

In parallel the voltage is the same and the reciprocals add up. $R_\text{eq}$ is always smaller than the smallest resistance; two equal resistors $R$ give $R_\text{eq} = R/2$.

Links

Topics: Circuiti elettrici Concepts: Resistenze in serie e parallelo

Related exercises: Esercizio svolto — rete mista serie e parallelo · Problema — vero o falso su serie e parallelo · Problema — resistenza equivalente serie-parallelo