Circuit Command

The Laws of Kirchhoff & Multi-Loop Analysis

AP Physics C: E&M

Unit 3: DC Circuits

The Core Principle

In AP Physics C, we transition from simple series-parallel combinations to complex networks that cannot be simplified using equivalent resistance alone. To conquer these, we rely on two fundamental conservation laws known as Kirchhoff's Laws.

While Ohm's Law (\(V = IR\)) describes the behavior of individual components, Kirchhoff's Laws describe the topology of the entire system. They are the circuit equivalent of Newton's Laws for mechanics.

Learning Objectives

01. Relate the Junction Rule to the Conservation of Charge.
02. Relate the Loop Rule to the Conservation of Energy.
03. Apply sign conventions correctly when traversing multi-loop circuits.
04. Construct and solve systems of linear equations for unknown currents.

1. The Junction Rule (KCL)

Kirchhoff’s Current Law (KCL), or the Junction Rule, is a direct consequence of the Conservation of Charge. In a steady-state circuit, charge does not "pile up" at any point.

Formal Definition

\[ \sum I_{in} = \sum I_{out} \]

"The sum of all currents entering a junction must equal the sum of all currents leaving that junction."

I₁

I₂

I₃

For the diagram above: \( I_1 = I_2 + I_3 \)

2. The Loop Rule (KVL)

Kirchhoff’s Voltage Law (KVL), or the Loop Rule, is a statement of the Conservation of Energy. Since the electrostatic force is conservative, the work done in moving a charge around any closed path must be zero.

Formal Definition

\[ \sum \Delta V_{loop} = 0 \]

"The algebraic sum of the changes in potential around any closed loop must be zero."

Crucial: Sign Conventions

Resistors (R)

With Current

\(-IR\)

Against Current

\(+IR\)

EMF Sources (\(\mathcal{E}\))

- → +

Negative to Positive

\(+\mathcal{E}\)

+ → -

Positive to Negative

\(-\mathcal{E}\)

3. Strategic Analysis Workflow

1

Label the Circuit

Mark every unique branch with a current (\(I_1, I_2, ...\)) and a direction arrow.

2

Apply the Junction Rule

Write current balance equations for all but one junction.

3

Apply the Loop Rule

Write voltage sum equations for enough independent loops to cover all branches.

Equation Count Requirement

To find N unknown currents, you need N independent equations.

L1

L2

4. Anatomy of a Circuit Setup

Analyze this multi-loop blueprint. Notice that components are fully integrated into the wire paths. Standard symbols: Zigzag for resistors and parallel plates for batteries.

Junction Equation

\( I_1 + I_3 = I_2 \)

Loop Rule Equations

Left Loop (CW): \( +\mathcal{E}_1 - I_1 R_1 - I_2 R_2 = 0 \)

Right Loop (CCW): \( +\mathcal{E}_2 - I_3 R_3 - I_2 R_2 = 0 \)

5. Worked Example: Numerical Model

Equations:

Node: \( I_1 + I_3 = I_2 \)

L1: \( 12 - 4I_1 - 2I_2 = 0 \)

L2: \( 6 - 8I_3 - 2I_2 = 0 \)

Final solved currents: \(I_1 = 1.93A, I_2 = 2.14A, I_3 = 0.21A\). All directions match predictions.

?

Self-Study Practice

Problem 1: Junction Rule

Five wires meet. \(I_1=2A\) (in), \(I_2=5A\) (out), \(I_3=1A\) (in), \(I_4=4A\) (in). Find magnitude and direction of the fifth wire.

Problem 2: Loop Basics

A loop has a 20V battery, 10Ω resistor, and 15V battery (opposing). Calculate current.

Problem 3: Multi-Loop Mastery

Find the current in each branch for the circuit shown. Label nodes and loops before solving.

Problem 4: Conceptual Depth

A multi-loop analysis yields: \(I_1 = 1.5A, I_2 = -0.5A, I_3 = 1.0A\).

a) Explain the negative sign for \(I_2\).

b) Calculate power dissipated in a \(20\Omega\) branch with \(I_3\).

c) If a 12V battery is in the \(I_1\) branch and current enters its positive terminal, is it charging or supplying? Calculate the rate.

Kirchhoff Command Summary

The Rules

Junction: \(\sum I = 0\)

Loop: \(\sum V = 0\)

Sign Checklist

• With \(I\) over \(R\): \(-IR\)
• Against \(I\) over \(R\): \(+IR\)
• Battery \((- \to +)\): \(+\mathcal{E}\)
• Battery \((+ \to -)\): \(-\mathcal{E}\)

AP PHYSICS C MASTER // CIRCUITRY UNIT

Circuit Command Reading v5

Circuit Command Reading v5

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