Unit 11: Electric Circuits — Differential Equations and Kirchhoff Analysis

Electric Circuits in AP Physics C E&M

Unit 11 extends electrostatics into dynamic systems where charge flows continuously. The distinguishing feature of AP Physics C E&M circuit analysis is the treatment of RC circuits using first-order differential equations — a significant step beyond the steady-state analysis of introductory physics courses. Students must set up, solve, and interpret exponential solutions with meaningful time constants.

Current, Resistance, and Ohm's Law

Current is defined as I = dq/dt — the instantaneous rate of charge flow. This calculus definition becomes critical in RC circuit analysis where charge and current are functions of time. Resistance R = ρL/A connects bulk material properties to circuit behaviour. Ohm's law V = IR applies to resistive elements, and power dissipation P = IV = I²R = V²/R must be applied in multi-element circuits.

Kirchhoff's Laws

Kirchhoff's Current Law (KCL)

At any junction in a circuit, the sum of currents entering equals the sum leaving: ΣI_in = ΣI_out. This is a statement of charge conservation. AP FRQs test KCL in multi-loop circuits where students must define current variables and write a system of equations at each node.

Kirchhoff's Voltage Law (KVL)

Around any closed loop, the sum of potential differences equals zero: ΣΔV = 0. Students must correctly account for the sign of each voltage drop (positive for resistors traversed in the current direction; positive or negative for EMF sources depending on traversal direction). AP FRQs require students to write and solve simultaneous KVL equations for circuits with multiple loops.

RC Circuits and Differential Equations

Charging an RC Circuit

When a capacitor charges through a resistor from a battery (EMF = ε), applying KVL gives: ε − IR − q/C = 0. Substituting I = dq/dt yields the first-order linear ODE: R(dq/dt) + q/C = ε. The solution is q(t) = Cε(1 − e^(−t/RC)), with time constant τ = RC. Current is I(t) = dq/dt = (ε/R)e^(−t/RC). AP FRQs expect the full derivation including the separation of variables and integration step.

Discharging an RC Circuit

For a discharging capacitor: R(dq/dt) + q/C = 0. This separable ODE gives q(t) = Q₀e^(−t/RC), and I(t) = −(Q₀/RC)e^(−t/RC). The negative sign indicates current flows opposite to the charging direction. Students must show the separation of variables, ∫dq/q = −∫dt/(RC), leading to ln(q/Q₀) = −t/RC.

The Time Constant τ = RC

The time constant τ = RC has units of seconds and characterises the rate of charge/discharge. At t = τ, the capacitor has charged to (1 − 1/e) ≈ 63% of its final charge, or discharged to 1/e ≈ 37% of its initial charge. AP questions test both the physical interpretation of τ and its role in the exponential solution.

Key FRQ Skills for Unit 11

1. Writing the RC circuit ODE: Apply KVL to the circuit; substitute I = dq/dt; write the differential equation explicitly.
2. Solving by separation of variables: Separate dq and dt terms; integrate both sides with correct limits; exponentiate to obtain q(t).
3. Differentiating for current: Compute I(t) = dq/dt from the q(t) solution; interpret the sign and initial value.
4. Kirchhoff analysis: Write and solve simultaneous KCL and KVL equations for multi-loop resistive circuits.

Common Errors in Unit 11

• Writing I = q/t instead of I = dq/dt in the ODE setup
• Dropping the homogeneous solution when solving the charging ODE, giving q(t) = Cε instead of Cε(1 − e^(−t/RC))
• Incorrect signs in KVL — traversing a resistor opposite to current direction should increase potential
• Confusing τ = RC (time constant) with the half-life t₁/₂ = RC·ln2

Practising Unit 11

The Unit 11 AP-style test includes multi-loop Kirchhoff problems and full RC circuit ODE derivations. In every FRQ, write the differential equation explicitly before solving — AP scoring rubrics award points for the ODE setup independently of the final solution.