AP Physics C: E&M Unit 11 ReviewElectric Circuits

AP Physics C: E&M Unit 11 covers electric circuits, which is how charge actually moves and delivers energy in real devices. You'll analyze current, resistance, power, and multi-loop DC circuits, then finish with RC circuits where charge and current change exponentially over time. The single biggest idea is that Kirchhoff's rules are just conservation of energy and conservation of charge written in circuit language, so every circuit problem is secretly a conservation problem. This unit makes up 15-25% of the AP exam.

What this unit covers

Current and what's actually moving

• Current is the rate that charge passes through a cross section of wire, $I = dq/dt$. It's a flow rate, like liters per second but for coulombs.
• Microscopically, current comes from charge carriers drifting through the conductor at an average drift velocity, captured by $I = nqv_dA$. The drift speed is tiny; the field that pushes the charges propagates fast.
• Charge moves because of a potential difference. A battery's emf ($\mathcal{E}$) is the potential difference it would supply if it were ideal, with no internal resistance.
• Circuits can be closed (charge flows), open (a break stops flow), or shorted (charge flows through a path with no potential difference, which is usually a problem).

Resistance, Ohm's law, and power

• Resistance measures how strongly an object opposes charge flow. For a uniform resistor, $R = \rho \ell / A$. Longer means more resistance, thicker means less, and resistivity $\rho$ is the material's own property set by its atomic structure.
• Ohm's law, $I = \Delta V / R$, relates current, potential difference, and resistance. Ohmic materials have constant resistance at any current, so their I-V graph is a straight line through the origin. Non-ohmic elements (like a lightbulb filament that heats up) don't.
• Power is the rate energy is transferred or dissipated by an element, $P = I\Delta V$, with the derived forms $P = I^2R = \Delta V^2/R$. Pick the form that uses what you know.
• Bulb brightness tracks power. A huge fraction of conceptual circuit questions reduce to "which bulb dissipates more power after the switch closes?"

Compound DC circuits and measurement

• Series elements carry the same current (one path, no choices). Parallel elements share the same potential difference (the two ends of each branch are the same two points).
• Series resistors add, $R_{eq} = \sum R_i$. Parallel resistors add as inverses, $1/R_{eq} = \sum 1/R_i$, so the equivalent is always smaller than the smallest branch.
• Real batteries have internal resistance $r$. The terminal voltage is the emf minus the drop inside the battery, $\Delta V = \mathcal{E} - Ir$, so a battery delivers less voltage the harder you draw current from it.
• Ideal wires have negligible resistance, but you can only ignore wire resistance when the circuit has other elements that do have resistance.
• Ammeters measure current and go in series with zero ideal resistance, so they don't choke the current they're measuring. Voltmeters measure potential difference and go in parallel with infinite ideal resistance, so they don't steal current from the branch.

Kirchhoff's rules

• The loop rule says the sum of potential differences around any closed loop is zero, $\sum \Delta V = 0$. This is conservation of energy. A charge that travels a full loop ends at the same potential it started at.
• The junction rule says total current into a junction equals total current out, $\sum I_{in} = \sum I_{out}$. This is conservation of charge. Charge doesn't pile up at a node or vanish.
• Together they handle multi-loop circuits that series-parallel reduction can't simplify, like circuits with two batteries in different branches. Assign currents, write loop and junction equations, solve the system.

RC circuits and time-dependent behavior

• Capacitors combine opposite to resistors. In series, $1/C_{eq} = \sum 1/C_i$ (equivalent is smaller than any individual). In parallel, $C_{eq} = \sum C_i$.
• Applying the loop rule to a charging RC circuit gives a differential equation, $\mathcal{E} = R\frac{dq}{dt} + \frac{q}{C}$. Solving it gives exponential charging and discharging.
• The time constant $\tau = R_{eq}C_{eq}$ sets how fast the capacitor charges or discharges. After one time constant, a charging capacitor reaches about 63% of its final charge; a discharging one falls to about 37%.
• Limiting behavior is the fastest conceptual tool. At $t = 0$, an uncharged capacitor acts like a bare wire. As $t \to \infty$, a fully charged capacitor acts like an open switch and its branch carries no current.

Unit 11, Electric Circuits at a glance

Why Unit 11, Electric Circuits matters in AP Physics E&M

This unit is where the field and potential machinery from earlier units finally produces something measurable. Potential difference stops being an abstract scalar and becomes the thing that drives current through your phone charger. It is also the unit where AP Physics C expects calculus to show up in a circuit context for the first time.

• Kirchhoff's rules tie circuits directly to the course's two deepest conservation laws. The loop rule is energy conservation; the junction rule is charge conservation.
• RC circuits introduce the first differential equation of the E&M course. Setting up and solving $\mathcal{E} = R\,dq/dt + q/C$ is exactly the skill you reuse later for inductors.
• Power dissipation connects circuits back to energy transfer, the throughline of all of physics. Every $I^2R$ term is electrical energy becoming thermal energy.

How this unit connects across the course

• Potential difference, the engine of every circuit, is defined in Electric Potential (Unit 9). The loop rule is just "potential is path-independent around a closed loop" turned into an equation.
• Capacitance, dielectric behavior, and stored energy $U = \frac{1}{2}C\Delta V^2$ come from Conductors and Capacitors (Unit 10). RC circuits put those capacitors into loops and make them charge over time.
• The conduction model of charge carriers in conductors builds on Electric Charges, Fields, and Gauss's Law (Unit 8), where you learned how charge behaves in and on conducting materials.
• The currents you compute here create magnetic fields in Magnetic Fields and Electromagnetism (Unit 12), and the RC differential equation method is reused almost line for line on LR circuits in Electromagnetic Induction (Unit 13).

Key equations and processes

• $I = dq/dt$ defines current as the rate of charge flow; differentiate $q(t)$ or integrate $I(t)$ to convert between charge and current.
• $I = nqv_dA$ connects macroscopic current to carrier density, charge, drift velocity, and cross-sectional area.
• $R = \rho\ell/A$ gives resistance from geometry and material; use it when a wire is stretched, cut, or compared across materials.
• $I = \Delta V / R$ (Ohm's law) relates the three basic circuit quantities for ohmic elements.
• $P = I\Delta V = I^2R = \Delta V^2/R$ gives power dissipated or delivered; the brightness of a bulb scales with its power.
• $R_{eq} = \sum R_i$ (series) and $1/R_{eq} = \sum 1/R_i$ (parallel) collapse resistor networks into a single equivalent.
• $1/C_{eq} = \sum 1/C_i$ (series) and $C_{eq} = \sum C_i$ (parallel) do the same for capacitors, with the rules flipped relative to resistors.
• $\Delta V_{terminal} = \mathcal{E} - Ir$ accounts for a battery's internal resistance.
• $\sum \Delta V = 0$ (loop rule) and $\sum I_{in} = \sum I_{out}$ (junction rule) generate the system of equations for any multi-loop circuit.
• $\mathcal{E} = R\frac{dq}{dt} + \frac{q}{C}$ is the RC charging equation from the loop rule; solve it to get exponential $q(t)$ and $I(t)$, with time constant $\tau = R_{eq}C_{eq}$.

Unit 11, Electric Circuits on the AP exam

Electric Circuits is worth 15-25% of the exam, which makes it one of the heaviest-weighted units in AP Physics C: E&M. On the multiple-choice section, expect ranking tasks (order bulbs by brightness, rank currents through resistors), switch problems (what changes the instant a switch closes versus a long time later), and quick equivalent-resistance or equivalent-capacitance calculations.

On the free-response section, circuits questions typically ask you to derive expressions symbolically, not just plug in numbers. Common moves include applying the loop and junction rules to a multi-loop circuit with internal resistance, deriving the RC differential equation from the loop rule and solving it for $q(t)$ or $I(t)$, and analyzing limiting cases at $t = 0$ and $t \to \infty$. Lab-style prompts also show up, such as designing a setup with ammeters and voltmeters placed correctly, or linearizing data (plotting resistance versus $\ell/A$ to extract resistivity from a slope, or using a log plot to find a time constant). Practice writing clean derivations where every step is justified by a named principle, because that is exactly what the rubric rewards.

Essential questions

• How do conservation of energy and conservation of charge fully determine the behavior of any circuit?
• Why does a battery's usable voltage drop when you connect it to a circuit that draws more current?
• What makes the behavior of an RC circuit time-dependent, and why does it follow an exponential curve instead of a linear one?
• How can the same set of resistors deliver completely different power depending on how they're wired together?

Key terms to know

• Electric current: The rate at which charge passes through a cross-sectional area of a conductor, $I = dq/dt$.
• Drift velocity: The average velocity of charge carriers moving through a conductor, much slower than the signal speed.
• Electromotive force (emf): The potential difference a battery would supply if it had no internal resistance.
• Resistivity: A material property quantifying how strongly the material opposes charge flow, independent of the object's shape.
• Ohmic material: A material whose resistance stays constant at any current, producing a linear I-V graph.
• Internal resistance: The resistance inside a real battery that lowers terminal voltage as current increases.
• Series connection: A connection where charge has exactly one path, so every element carries the same current.
• Parallel connection: A connection where charge has multiple paths, so every branch has the same potential difference.
• Equivalent resistance: The single resistance that could replace a network of resistors without changing the circuit's behavior.
• Ammeter: A meter wired in series to measure current, ideally with zero resistance.
• Voltmeter: A meter wired in parallel to measure potential difference, ideally with infinite resistance.
• Short circuit: A path where charge flows with no change in potential difference, bypassing other elements.
• Time constant ($\tau$): The quantity $R_{eq}C_{eq}$ that measures how quickly an RC circuit's capacitor charges or discharges.

Common mix-ups

• Series and parallel rules flip between resistors and capacitors. Series resistors add directly, but series capacitors add as inverses. If you memorize one set, the other is its mirror image.
• A voltmeter reads the potential difference across an element, not "the voltage at a point." It must connect to two points, in parallel with the element. Wiring a voltmeter in series effectively opens the circuit.
• The capacitor in an RC circuit behaves like a wire at $t = 0$ (uncharged, no potential difference across it) and like an open switch as $t \to \infty$ (fully charged, no current through its branch). Mixing these up reverses every limiting-case answer.
• Terminal voltage is not emf. When current flows, the battery's terminals show $\mathcal{E} - Ir$, which is why a "12 V" battery reads less than 12 V under load.