Equivalent Resistance of a Network

A resistor network looks complicated, but the battery driving it doesn't care about the maze inside — it only feels total opposition to current. That's the equivalent resistance: one number that reproduces the same relationship between the voltage applied and the current drawn. Finding it means spotting simple patterns hiding inside the mess: resistors in series, where the same current must pass through each in turn, and resistors in parallel, where the same voltage splits the current across multiple paths.

Work from the inside out. Find two resistors that are purely in series (R_eq = R1 + R2) or purely in parallel (1/R_eq = 1/R1 + 1/R2), replace them with their single equivalent, redraw the simplified circuit, and repeat. Each combination shrinks the network by one resistor until a single value remains. Some networks, like a cube of twelve equal resistors between opposite corners, have no resistor that's purely series or purely parallel at first glance — there you use symmetry to see which junctions sit at the same potential, which lets you merge nodes and then collapse the simplified network the same way.

This collapsing technique is the algebra behind everything else in steady-state circuit analysis: once you have one equivalent resistance, Ohm's law gives you total current immediately, and you can work backward to find voltage and current at each original resistor. It also sets up Kirchhoff's rules, which you need when a network can't be reduced by series and parallel combinations alone, such as circuits with multiple batteries or bridge configurations.