Resistors in Parallel (Derivation)

1. Introduction: Many Paths, One Shared Push

You already know a few things from earlier in this chapter. Voltage $V$ is the push across a part, measured in volts (V). Current $I$ is the flow through it, measured in amperes (A). And resistance $R$, measured in ohms (Ω), is the one number for how hard a part fights that flow. Ohm's law ties them together: $V = IR$.

You have also just seen resistors in series — joined one after another on a single road. Now we change the wiring. Suppose you connect two or three resistors side by side, each one bridging the same two points, so the current has several roads to choose from. The whole group still has some total resistance. What single resistor could you swap in to replace the whole group and change nothing?

That single replacement is called the equivalent resistance — the one resistor that behaves exactly like the whole group put together. For resistors in parallel, we are going to prove, step by step, that this equivalent resistance $R_p$ obeys:

$\dfrac{1}{R_p} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3}$

By the end of this lesson you will not just remember that formula. You will be able to build it yourself from two simple facts, see why the equivalent resistance comes out smaller than any single resistor, and use it to find currents in a real circuit — the way your home is actually wired.