Resistors in Series

Consider 2 resistors in series, as shown in Figure 1. Imagine that there is a current flowing through the wire.

The current at any point along the wire must be the same. Thus, we can call it $I$. Suppose the current is flowing to the right, i.e. from $A$ to $B$. Let point $C$ be between the resistors $R_1$ and $R_2$.

The potential difference between $A$ and $B$ is just $I(R_1 + R_2)$. Now, if we were to take two more points, and make the potential difference $I(R_1 + R_2)$, and run the same current $I$ between them, we would want a resistor with resistance $R = R_1 + R_2$ between $A$ and $B$. That way, $\Delta V = IR = I(R_1 + R_2)$.

The notion of putting one resistor in place of two or more resistors in series is called equivalent resistance. It helps a lot in problems where you have a ton of resistors and you have to figure out the potential difference between them. In this case, if you have $n$ resistors, $R_1$ through $R_n$, connected in series, then the equivalent resistor that you can put in the place of all those resistors would just be one with resistance $R_1 + R_2 + \cdots + R_n$.

Let's do one quick example before we close:

Example: Suppose that, in Figure 1, $R_1 = 1\ \Omega$ and $R_2 = 2\ \Omega$. The voltage difference between $A$ and $B$ is $6\ \text{V}$. What is the voltage difference across $R_2$?

That's it!