AP Physics 2: DC Circuits

Direct-current (DC) circuits are one of the most practical topics in AP Physics 2 because they connect clean mathematical rules to real electrical behavior. Whether you are analyzing a simple flashlight circuit or a more complex network with multiple loops, the core ideas stay the same: charge flows because of electric potential differences, circuit elements limit or store energy, and conservation laws govern everything.

This article covers the essential tools for DC circuit analysis: current and resistance, Ohm’s law, series and parallel combinations, Kirchhoff’s rules, power dissipation, and the time-dependent behavior of RC circuits.

Foundations: Charge Flow and Potential Difference

Current and conventional direction

Electric current $I$ is the rate of flow of charge through a cross-section of a conductor:

$I=\frac{\Delta Q}{\Delta t}$

The SI unit is the ampere (A), equal to coulombs per second. In circuit diagrams, current is shown in the conventional direction, from higher potential to lower potential, even though electrons in metal wires drift in the opposite direction.

Resistance and what it means physically

Resistance $R$ measures how strongly a component opposes current for a given potential difference. In many AP problems, resistors are treated as ideal: their resistance is constant and independent of voltage, current, and temperature. Real materials can deviate from this, but the ideal model is accurate for typical circuit analysis.

Ohm’s Law and Circuit Elements

Ohm’s law

For an ohmic resistor, the relationship between potential difference and current is linear:

$V=IR$

This is not a universal law for all devices. Components like diodes and filament bulbs can be non-ohmic, but AP Physics 2 DC circuit problems typically focus on ohmic resistors unless stated otherwise.

Batteries, emf, and internal resistance

A battery provides energy per unit charge, represented by its electromotive force (emf) $\mathcal{E}$. In idealized problems, the battery maintains a fixed potential difference equal to $\mathcal{E}$ across its terminals.

More realistic models include internal resistance $r$, which causes the terminal voltage to drop when current is drawn:

$V_{\text{terminal}}=\mathcal{E}-Ir$

This is a common source of conceptual mistakes: emf is not automatically the same as the measured terminal voltage under load.

Resistor Combinations: Series and Parallel

Reducing a circuit to an equivalent resistance is often the fastest route to current and voltage values.

Series resistors

Resistors in series carry the same current. The equivalent resistance is:

$R_{\text{eq}}=R_1+R_2+\dots$

The voltage divides across the resistors in proportion to their resistances. This is the basis of a voltage divider, where a chosen resistor ratio produces a desired fraction of the source voltage.

Parallel resistors

Resistors in parallel share the same voltage across them. The total current splits among branches, and the equivalent resistance satisfies:

$\frac{1}{R_{\text{eq}}}=\frac{1}{R_1}+\frac{1}{R_2}+\dots$

A useful reality check: adding a parallel branch always decreases the equivalent resistance because it adds an additional path for current.

Mixed networks

Most real problems involve networks that are neither purely series nor purely parallel. The strategy is to simplify step-by-step where possible, then use Kirchhoff’s rules when simplification stalls.

Kirchhoff’s Rules: The Core of Circuit Analysis

When a circuit has multiple loops and junctions, Kirchhoff’s rules provide a systematic method based on conservation laws.

Kirchhoff’s junction rule (KCL)

At any junction, the sum of currents entering equals the sum of currents leaving:

$\sum I_{\text{in}}=\sum I_{\text{out}}$

This is conservation of charge. Charge does not “pile up” at a junction in steady-state DC conditions.

Kirchhoff’s loop rule (KVL)

Around any closed loop, the algebraic sum of potential changes is zero:

$\sum \Delta V=0$

This reflects conservation of energy. A charge completing a loop returns to its starting potential, so the net change must cancel.

Sign conventions that prevent errors

Kirchhoff problems are won or lost on sign consistency:

• For a resistor, moving in the direction of current gives a drop $-IR$; moving opposite current gives a rise $+IR$.
• For a battery, moving from the negative to the positive terminal gives a rise $+\mathcal{E}$; moving from positive to negative gives a drop $-\mathcal{E}$.

Pick loop directions and assumed current directions freely, then let the algebra determine whether an assumed current is negative.

Power Dissipation and Energy in DC Circuits

Power is the rate of energy transfer. In circuits, resistors convert electrical energy into thermal energy (and sometimes light, sound, etc.).

Power formulas

The basic definition is:

$P=IV$

Using Ohm’s law, two equivalent forms are often convenient:

$P=I^{2}R$, and $P=\frac{V^2}{R}$

These are not three different ideas, just different ways to compute the same physical quantity depending on what you know.

Practical interpretations

• In series circuits, the same current flows through each resistor, so power scales with resistance: $P\propto R$ (using $P=I^{2}R$).
• In parallel circuits, the same voltage is across each branch, so power decreases as resistance increases: $P\propto 1/R$ (using $P=V^2/R$).

This explains why low-resistance paths in parallel can draw large currents and dissipate significant power.

Capacitors in DC Circuits: RC Charging and Discharging

Capacitors store energy in an electric field and introduce time dependence into otherwise steady circuits.

Key capacitor relationships

Capacitance $C$ relates charge and voltage:

$Q=CV$

Energy stored in a capacitor is:

$U=\frac{1}{2}C{V}^2$

What happens in steady-state DC

In a DC circuit, after a long time:

• A capacitor behaves like an open circuit (no steady current through it).
• The capacitor’s voltage becomes constant.
• Currents elsewhere in the circuit settle into steady values.

This is a major conceptual anchor: capacitors do allow current while charging or discharging, but not indefinitely in a simple DC source-resistor-capacitor setup.

The RC time constant

For a series resistor and capacitor, the characteristic time scale is:

$\tau =RC$

The time constant $\tau$ is the time for the capacitor voltage (or charge) to move about 63% of the way from its initial value toward its final value during charging, or to drop to about 37% during discharging.

Charging behavior

For a capacitor charging through a resistor from a battery $\mathcal{E}$:

• Capacitor voltage:

$V_C(t)=\mathcal{E}\left(1-e^{-t/RC}\right)$

• Current:

$I(t)=\frac{\mathcal{E}}{R}{e}^{-t/RC}$

At $t=0$, the capacitor initially behaves like a wire (maximum current). As time increases, the current decreases because the growing capacitor voltage reduces the voltage across the resistor.

Discharging behavior

If a charged capacitor discharges through a resistor:

• Capacitor voltage:

$V_C(t)=V_0{e}^{-t/RC}$

• Current magnitude:

$I(t)=\frac{V_0}{R}{e}^{-t/RC}$

The exponential decay reflects energy leaving the capacitor and being dissipated by the resistor.

Problem-Solving Approach That Works Under Pressure

1. Draw and label: Mark currents, polarities across resistors, and battery terminals.
2. Simplify where possible: Combine series and parallel resistors before using Kirchhoff.
3. Apply KCL at junctions: Write current relationships first, especially in multi-branch circuits.
4. Apply KVL to independent loops: Write one equation per loop, using consistent sign conventions.
5. Check units and limits: Equivalent resistance should make sense (series increases, parallel decreases). In RC problems, verify behavior at $t=0$ and as $t\to \infty$.

What AP Physics 2 Emphasizes

AP Physics 2 DC circuits are less about memorizing formulas and more about interpreting physical meaning: why junction currents must balance, why loop voltages must sum to zero, and how energy is conserved as electrical potential energy becomes thermal energy. Mastering Kirchhoff’s rules and the exponential behavior of RC circuits gives you the tools to analyze nearly any DC network you will encounter in the course.