Summary

General Strategy

Model Assume that wires and, where appropriate, batteries are ideal.

Visualize Draw a circuit diagram. Label all quantities.

Solve Base the solution on Kirchhoff’s laws.

Review Verify that

The sum of potential differences across series resistors matches $Δ V$ for the equivalent resistor.
The sum of the currents through parallel resistors matches $I$ for the equivalent resistor.

For a closed loop:

For a junction:

A potential difference $Δ V$ between the ends of a conductor with resistance $R$ creates a current

I = \frac{Δ V}{R}

The energy used by a circuit is supplied by the emf $ε$ of the battery through the energy transformations

E_{chem} \to U \to K \to E_{th}

The battery supplies energy at the rate

P_{bat} = I ε

The resistors dissipate energy at the rate

P_{R} = I Δ V_{R} = I^{2} R = \frac{(Δ V_{R})^{2}}{R}

Groups of resistors can often be reduced to one equivalent resistor.

R_{eq} = R_{1} + R_{2} + R_{3} + \dots

R_{eq} = {(\frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + \dots)}^{- 1}

The charge on and current through a discharging capacitor are

\begin{array}{l} Q = Q_{0} e^{- t / τ} \\ I = - \frac{d Q}{d t} = \frac{Q_{0}}{τ} e^{- t / τ} = I_{0} e^{- t / τ} \end{array}

where $τ = R C$ is the time constant.