Electric Circuits and Kirchhoff's Rules

Circuits are one of the most predictable topics in AP Physics 2. Nearly every question is solved with three ideas: equivalent resistance, Kirchhoff's two rules, and the power formulas.

Ohm's law and power

V = IR

P = IV = I²R = V²/R

Choose the power form that uses the quantities you already know. For resistors in series the current is shared, so I²R is convenient; in parallel the voltage is shared, so V²/R is convenient.

Series and parallel

• Series: same current through each resistor. R_eq = R₁ + R₂ + ...
• Parallel: same voltage across each branch. 1/R_eq = 1/R₁ + 1/R₂ + ...

Adding a resistor in parallel always lowers the equivalent resistance, so the total current from an ideal battery goes up.

Kirchhoff's rules

1. Junction rule (charge conservation): the current into a junction equals the current out.
2. Loop rule (energy conservation): around any closed loop, the sum of potential changes is zero. Crossing a resistor in the direction of the current is a drop of IR; crossing a battery from − to + is a rise of ε.

Worked example

A 12 V ideal battery is connected to a 2 Ω resistor in series with a parallel pair of 6 Ω and 3 Ω.

1. The parallel pair: 1/R = 1/6 + 1/3 = 1/2, so 2 Ω.
2. Total resistance: 2 + 2 = 4 Ω, so the battery current is 12/4 = 3 A.
3. Voltage across the 2 Ω resistor: 3 × 2 = 6 V, which leaves 6 V across the parallel pair.
4. Branch currents: 6/6 = 1 A and 6/3 = 2 A. Check with the junction rule: 1 + 2 = 3 A.
5. Power in the 3 Ω resistor: 6²/3 = 12 W.

Capacitors in circuits

• Just after the switch closes: an uncharged capacitor behaves like a wire. The current is as large as the rest of the circuit allows.
• After a long time: a fully charged capacitor behaves like a break in the wire. No current flows in its branch, and its voltage equals whatever it is in parallel with.
• Between the two: the charge changes gradually, on a time scale set by RC.

Bulbs and brightness

Brightness follows power. With an ideal battery, adding a second identical bulb in parallel leaves the first bulb's voltage, and so its brightness, unchanged, while the battery supplies twice the current. Adding it in series halves the current, and each bulb becomes dimmer.

The free response approach

Redraw the circuit if it is messy, write the rule you are using in words, and when asked to rank or compare, justify with a relation such as P = V²/R rather than with intuition.