Simple Circuits

The study of electric circuits grew from centuries of inquiry into the nature of electricity. Early investigators observed static sparks and the attraction of amber to light objects, but a systematic understanding of steady current flow only became possible once chemical batteries could supply a continuous source of charge. The mathematical relationships that govern simple circuits—linking voltage, current, and resistance—were forged through careful experimentation in the early nineteenth century, and they remain the backbone of every electrical system today, from smartphone chargers to hospital defibrillators.

The central question these pioneers answered is deceptively simple: How does the arrangement of components in a circuit determine the current through each element and the voltage across it? Answering that question requires understanding series and parallel topologies, the conservation laws that constrain charge and energy flow, and the mathematical tools—Ohm's law and Kirchhoff's rules—that translate physical principles into solvable equations.

A simple circuit is any closed conducting path through which charge can flow, driven by a source of electromotive force (emf) such as a battery. The behavior of every simple circuit rests on a handful of foundational ideas that connect the macroscopic quantities of voltage, current, and resistance to the microscopic motion of charge carriers.

The diagram above captures the two fundamental ways to arrange resistors. In a series configuration, there is only one path for charge to follow, so the current I is identical through every element. Ohm's law applied to each resistor gives individual voltage drops V₁ = IR₁, V₂ = IR₂, and V₃ = IR₃, which must sum to the battery emf by Kirchhoff's loop rule. In a parallel configuration, multiple paths exist between the same two nodes, so each branch experiences the full battery voltage. The junction rule then requires the total current entering a node to equal the sum of branch currents, I₁ + I₂. These two patterns form the building blocks for analyzing any circuit, no matter how complex.

During discharge, internal resistance makes terminal voltage smaller than the emf. The source power divides between useful load power and internal heating.

During discharge, internal resistance makes terminal voltage smaller than the emf. The source power divides between useful load power and internal heating.

The loop rule (Kirchhoff's voltage law) is a statement of conservation of energy: as you traverse any closed loop, the algebraic sum of all potential gains (from batteries) and drops (across resistors) equals zero. The junction rule (Kirchhoff's current law) is a statement of conservation of charge: current entering a node must equal current leaving it, because charge cannot accumulate in steady state. Together these two rules, combined with Ohm's law, provide enough independent equations to solve for every unknown current and voltage in any resistor network.

Many real circuits contain combination (series-parallel) networks. The strategy for analyzing them is to simplify from the inside out: identify groups of resistors that are purely in series or purely in parallel, replace each group with its equivalent resistance, and repeat until the entire network reduces to a single equivalent resistor connected to the battery. Once you know the total current from the battery, you can 'unpack' the reductions in reverse order to find the current through and voltage across every individual resistor.

Simple resistor circuits form the conceptual scaffolding for more complex topics you will encounter in AP Physics 2 and beyond. The same Kirchhoff laws that let you analyze a two-resistor loop also govern circuits containing capacitors (RC circuits), where current is not constant but decays exponentially as the capacitor charges. Internal resistance of real batteries, explored later in the course, adds a series resistor inside the battery itself, causing the terminal voltage to sag under load. Understanding simple DC circuits also prepares you for AC circuit analysis in more advanced courses, where impedance replaces resistance and phasors replace scalar voltages.

On the AP Physics 2 exam, circuit problems frequently ask you to predict what happens when a switch is opened or closed, a resistor is added or removed, or a battery's internal resistance is non-negligible. Mastering the simple circuit principles in this lesson—particularly the conservation-based reasoning encoded in Kirchhoff's rules—gives you the analytical toolkit needed for those more nuanced scenarios.