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Compound Direct Current Circuits

AP Course

AP Physics C: Electricity and Magnetism

Updated for the AP Physics C: Electricity and Magnetism framework effective fall 2024, with calculus-based fields, circuits, and induction.

Build calculus-based E&M with original field models, derivations, experiments, quizzes, and practice sets.

AP Physics C: Electricity and Magnetism · Unit 11 · Topic 11.5

Compound Direct Current Circuits

Differential equations and conservation rules predict steady and transient behavior in resistive-capacitive networks.

Experimental relationship · Compound Direct Current CircuitsUse the overall pattern, spread, uncertainty, and possible outliers when evaluating the proposed relationship.

1. Topic Lens

Compound Direct Current Circuits is studied through electric circuits. Connect the system boundary, interacting parts, and measurable evidence before applying a formula.

\[I=\frac{dQ}{dt}\]

2. Why the Formula Works

The relationship is built from definitions and conservation reasoning:

  1. For a charging loop, Kirchhoff's rule gives ℰ-IR-Q/C=0.
  2. Use I=dQ/dt to form R dQ/dt+Q/C=ℰ.
  3. Solving the first-order equation with Q(0)=0 gives Q=Cℰ(1-e^{-t/RC}) and V=Q/C.
\[V_C(t)=\mathcal E\left(1-e^{-t/RC}\right)\]

3. Detailed Visual Model

Compound Direct Current CircuitsTopic-matched fine pixel-art model · use the lesson's equations and labeled diagrams for exact scientific meaning.

4. Worked Example and Lab Link

For R=2 MΩ and C=3 μF, find the time constant.

Answer: τ=RC=(2×10⁶)(3×10⁻⁶)=6 s.

Investigation idea: Collect capacitor voltage versus time, linearize the exponential, and estimate R or C with uncertainty.

Common trap: At one time constant a charging capacitor reaches about 63%, not 100%, of its final voltage.

Checkpoint · Topic 11.5

Explain how compound direct current circuits supports or limits this conclusion: τ=RC=(2×10⁶)(3×10⁻⁶)=6 s.

Official curriculum reference: College Board AP Physics C: Electricity and Magnetism course page. The explanation and worked example are independently written for this study site.