Equivalent Resistance

In Resistors we defined the resistance of anything as $R = Δ V / I$ : we apply a potential difference to two different places on the object (called terminals) and measure the current that flows. This is even true for combinations of resistors. The resistance of a set of resistors is called that set's equivalent resistance $R_{e q}$ :

R_{e q} = \frac{Δ V}{I}

We can find the equivalent resistance of any set of resistors by attaching them to a battery and solving the circuit to find the current in and out of the set. But there are some configurations of resistors where the equivalent resistance is easily found:

Resistors in Series

Two or more circuit elements are in series if the current $I$ through each of them is exactly the same. If you can trace a path through all the elements without crossing a junction (where the current changes), then they are in series.

For resistors in series, the potential drop across each one is $Δ V_{1} = I R_{1}$ etc, and the total drop across all the resistors is the sum of the individual drops. Thus $\begin{aligned} Δ V_{t o t} & = Δ V_{1} + Δ V_{2} + \dots \\ = I R_{1} + I R_{2} + \dots \\ = I (R_{1} + R_{2} + \dots) \end{aligned}$ Notice we use the same $I$ for all the resistors because the current is the same. This is also the current into and out of the set of resistors, and so the total potential drop is $Δ V_{t o t} = I R_{e q}$ , and so $I R_{e q} = I (R_{1} + R_{2} + \dots)$ Dividing both sides by $I$ gives us

R_{e q} = R_{1} + R_{2} + \dots

for resistors in series

Resistors in Parallel

Two or more circuit elements are in parallel if the potential drop $Δ V$ across each of them is exactly the same. For instance, in the three examples here, the red wires are all at the same potential $V_{i}$ , while the blue wires are all at the same potential $V_{f}$ . Because the two resistors in the first two examples touch the red and the blue, the drop across them is $Δ V = V_{f} - V_{i}$ , and must be the same. In the third example, however, only onw resistor goes from red or blue, so those two resistors are not in parallel.

For parallel resistors, it is the currents that add together, not the potential drops, and we use the fact that $I_{1} = \frac{Δ V}{R_{1}}$ , and that $I_{t o t} = \frac{Δ V}{R_{e q}}$ : $\begin{aligned} I_{t o t} & = I_{1} + I_{2} + \dots \\ \frac{Δ V}{R_{e q}} & = \frac{Δ V}{R_{1}} + \frac{Δ V}{R_{2}} + \dots \end{aligned}$ If we divide both sides by $Δ V$ , we get

\frac{1}{R_{e q}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \dots

(for resistors in parallel)