Equivalent resistance

Also called Total resistance, R_eq

Equivalent resistance is the single resistance that could replace a group of resistors without changing anything the rest of the circuit sees. Resistors in series add directly; resistors in parallel add as reciprocals, so a parallel group resists less than any one of its members.

The CED licenses the whole idea at 11.5.A.2: a collection of resistors in a circuit may be analyzed as though it were a single resistor with an equivalent resistance $R_{\text{eq}}$ . Both the AP Physics 2 and the AP Physics C: E&M sheets print the two rules:

R_{\text{eq},s} = \sum_i R_i \qquad \frac{1}{R_{\text{eq},p}} = \sum_i \frac{1}{R_i}

Why that way round. In series, any charge passing through one element must proceed through all of them and has no other path, so the current is common and the potential differences add. In parallel, charges may take one of two or more paths and the potential difference across each path is the same, so the currents add. Add the thing that is not shared, and the rule follows.

The CED spells out the parallel consequence at 11.5.A.2.iii: adding a parallel branch increases the number of paths available to charges, so the equivalent resistance of the group decreases. Adding a resistor lowering the total resistance is the result students distrust; it is correct, and the reciprocal sum guarantees $R_{\text{eq},p}$ is smaller than the smallest branch.

These rules are the inverse of the ones for [equivalent capacitance](/glossary/equivalent-capacitance). Capacitors in series combine reciprocally and capacitors in parallel add, which is precisely the swap that costs marks under time pressure.

Reducing an actual network step by step is the series and parallel guide's job, and the junction rule is where the parallel formula comes from.

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