Resistors in Parallel

Suppose we have a circuit like that shown in Figure 1. There are 2 resistors, called $R_1$ and $R_2$.

For the rest of this "lecture," I will denote $V_{XY}$ as the potential difference between points $X$ and $Y$. (Of course, it really should be written as $\Delta V_{XY}$, but $\Delta$ takes 6 characters to write in $\LaTeX$, so it'll save me work. Remember that whenever you see $V$, it really means $\Delta V$.)

Remember that, in any circuit that we consider in physics problems, the wires are ideal (unless otherwise stated), which means that the resistance, and hence voltage difference, is 0.

Using similar reasoning, we get that $V_{DE} = V_{AF}$. Hence, $V_{BC} = V_{DE}$. Thus, for any $n$ resistors connected in parallel, the voltage drop across each of them is the same. (We'll define parallel more rigorously in a later "lecture.")

Now, assume all resistors obey Ohm's law. For simplicity, let $V_{AF} = V$.

Now, if you go and review the "lecture" on resistors in series, you'll find that there was an explanation about equivalent resistance.

We can easily extend this argument to show that the equivalent resistance for $n$ resistors in parallel is just $R_{eq} = \dfrac1{\frac1{R_1}+\frac1{R_2}+\cdots+\frac1{R_n}}$.

That's it for now. Later we will explore more complicated circuits and do a few examples. We will have to use both series and parallel equivalent resistance formulas to bring together a whole bunch of resistors together into one, then figure out the current, then expand the resistors one by one again using everything we have derived so far.