AP Physics C: E&M Unit 3 — Learning DC Circuit Analysis

Resistor

Circuit element that opposes current and converts electrical energy into other forms (often thermal energy).

Ohm’s Law (ohmic resistor)

Relationship between potential difference and current for an ohmic resistor: V = IR.

Equivalent Resistance (Req)

A single resistance value that can replace a network of resistors while producing the same overall current–voltage behavior at the terminals.

Series Connection

Components connected end-to-end with only one path for charge; the same current flows through each element.

Series Equivalent Resistance

For resistors in series, the equivalent resistance is the sum: $R_{eq} = R_1 + R_2 + \text{...}$.

Voltage Division in Series

In a series branch, the total voltage is shared among resistors; each drop is Vi = I Ri (drops are equal only if resistances are equal).

Parallel Connection

Components sharing the same two nodes, creating multiple current paths; the voltage across each branch is the same.

Parallel Equivalent Resistance

For resistors in parallel, reciprocals add: 1/Req = 1/R1 + 1/R2 + …

Parallel Qualitative Fact (Req size)

For parallel networks, $R_{eq}$ is always less than the smallest individual branch resistance.

Current Splitting in Parallel

Total current is the sum of branch currents, and lower-resistance branches draw more current (Ibranch = V/Rbranch).

Node (Junction)

A connection point where two or more circuit elements meet; used to identify parallel branches and apply Kirchhoff’s Junction Rule.

Series–Parallel Reduction Strategy

Identify pure series/parallel “chunks,” replace them with equivalent resistances, repeat until simplified, then work backward to find individual currents/voltages.

Current (I)

Charge flow rate, defined by $I = \frac{dq}{dt}$.

Potential Difference (Voltage)

Energy per charge between two points, $\triangle V = \frac{\triangle U}{q}$.

Kirchhoff’s Rules

Universal DC circuit-analysis rules based on conservation laws: Junction Rule (charge) and Loop Rule (energy).

Kirchhoff’s Junction Rule (KJR)

At a node in steady-state DC, sum of currents entering equals sum leaving: ΣIin = ΣIout (algebraic sum is zero with consistent signs).

Kirchhoff’s Loop Rule (KLR)

Around any closed loop, the algebraic sum of potential changes is zero: $\textstyle \textstyle \bigg( \triangle V = 0 \bigg)$.

Resistor Sign Convention in KLR

Traversing a resistor in the direction of assumed current gives a drop $-IR$; traversing opposite the assumed current gives a rise $+IR$.

Battery (emf) Sign Convention in KLR

Crossing an ideal battery from − to + is a rise +\text{ℰ}; from + to − is a drop -\text{ℰ}.

Negative Current Result (Kirchhoff interpretation)

If solving gives I < 0, the actual current direction is opposite the assumed direction (not a failure).

Electrical Power (P)

Rate of energy transfer: $P = \frac{dE}{dt}$; in circuits $P = IV$ (with consistent sign convention).

Resistor Power Formula (current form)

Using V = IR, power dissipated in a resistor can be written as P = I^2 R.

Resistor Power Formula (voltage form)

Using V = IR, power dissipated in a resistor can be written as P = \frac{V^2}{R} (V is the voltage across that resistor).

Power Supplied vs. Power Dissipated

In ideal DC circuits, sources deliver power (often P_{source} = I\text{ℰ}) and resistors dissipate it; total power supplied equals total power dissipated (a sign-check/consistency check).

Energy from Constant Power

If power is constant over time t, the energy transferred is E = Pt (e.g., heat produced in a resistor over time).