Simple Circuits

The study of electric circuits grew from a centuries-long effort to understand and harness the flow of charge. Before the late eighteenth century, static electricity was a laboratory curiosity—dramatic but impractical. The breakthrough came when experimenters discovered how to maintain a steady current, transforming electricity from a parlor trick into the backbone of modern technology. Each milestone below marked a conceptual leap that brought the physics of circuits closer to the rigorous, quantitative framework you will use on the AP exam.

These discoveries frame a central question that the rest of this lesson addresses: given a network of resistors and an ideal battery, how do we predict the current through—and voltage across—every element in the circuit? Answering that question with precision is the goal of simple circuit analysis.

Before solving any circuit, you need a firm grip on four foundational ideas. These concepts connect the macroscopic quantities you measure in a lab—voltage readings on a multimeter, current readings on an ammeter—to the underlying physics of charge flow and energy transfer.

The diagram above captures the two fundamental topologies you will encounter in simple circuits. In the series configuration on the left, charge has only one path from the high-potential terminal of the battery back to the low-potential terminal, so the current through R₁ and R₂ is identical. Applying Kirchhoff's loop rule around the single loop yields ε = IR₁ + IR₂ = I(R₁ + R₂), which immediately gives the series equivalent resistance R_eq = R₁ + R₂. In the parallel configuration on the right, both resistors connect directly across the battery, so each experiences the full EMF. The junction rule at the top node requires the total current to split: I = I₁ + I₂ = ε/R₁ + ε/R₂, leading to 1/R_eq = 1/R₁ + 1/R₂. Notice that adding resistors in series always increases the equivalent resistance, while adding resistors in parallel always decreases it.

The quantitative backbone of simple circuit analysis rests on Ohm's law and Kirchhoff's two rules. Combining these allows you to reduce any resistor network to a single equivalent resistance, find the total current, and then work backwards to determine individual branch currents and voltage drops.

Most AP problems present a combination circuit—a mixture of series and parallel connections. The strategy is to identify the innermost purely series or purely parallel group, replace it with its equivalent resistance, and repeat until the entire network collapses to a single resistor. After finding the total current from ε and R_eq, you "un-collapse" step by step to recover individual currents and voltages.

The reduction strategy illustrated above works for any network that is reducible—one where every pair of elements can be classified as purely series or purely parallel. Certain topologies (like a Wheatstone bridge with an unbalanced galvanometer branch) are not reducible and require full Kirchhoff loop/junction analysis. On the AP exam, however, most simple-circuit FRQs involve reducible networks, so mastering this collapse-and-expand technique is essential.

Consider a circuit with a 24 V battery (internal resistance r = 1 Ω) connected to three external resistors: R₁ = 5 Ω in series with the parallel combination of R₂ = 12 Ω and R₃ = 4 Ω. Find the total current, the terminal voltage, the power dissipated in R₃, and the voltage across R₂.

Simple DC resistor circuits are the gateway to a much richer landscape of circuit analysis. In the AP Physics C curriculum and in later coursework, you will encounter capacitors and inductors that store energy rather than dissipate it, leading to time-dependent (transient) behavior governed by differential equations. The table below maps the simple-circuit concepts you have learned to their more general counterparts.

On the AP Physics C exam, RC circuits are explicitly tested and form the next natural topic after mastering simple circuits. The key insight is that the resistor equations you have learned still apply at each instant—Kirchhoff's loop rule simply yields a first-order ODE when a capacitor is present, because the capacitor voltage depends on the integral of the current. A solid command of Ohm's law and Kirchhoff's rules is therefore prerequisite infrastructure for every circuit topic that follows.