Circuits

Scope of this lecture

This lecture introduces direct-current circuit analysis from the simplest resistor combinations to multi-loop networks that require systematic equations. The plan follows the progression suggested by the problem set: first identify structure, then reduce circuits when possible, and finally compute unknown currents and voltages in circuits that can no longer be simplified by inspection alone.

Main themes:

electric current, voltage, resistance, and power,
Ohm’s law as a local relation for a circuit element,
series and parallel resistor combinations,
equivalent resistance as a model simplification tool,
current and voltage division in simple networks,
Kirchhoff’s current law and Kirchhoff’s voltage law,
systematic analysis of junctions, loops, and bridge-like circuits,
interpretation of ammeter and voltmeter readings,
choosing an efficient solution strategy before calculating.

Lecture plan

1. Basic circuit quantities and conventions

define charge flow, current, potential difference, resistance, and power,
establish sign conventions for current direction and voltage drop,
explain ideal wires, ideal sources, and ideal measuring instruments.

2. Ohm’s law and the meaning of resistance

present

\[ V = IR \]
distinguish between element law and whole-circuit reasoning,
show how resistance controls current for a given applied voltage.

3. Series connections

identify when elements carry the same current,
derive the equivalent resistance rule for series combinations,
connect the rule to the first problems on direct resistor reduction.

4. Parallel connections

identify when elements share the same voltage,
derive the equivalent resistance rule for parallel combinations,
compare parallel behavior with series behavior physically and mathematically.

5. Mixed series-parallel networks

develop a reduction workflow for circuits that can be simplified step by step,
emphasize redrawing the circuit after each reduction,
prepare students for equivalent-resistance tasks like Problems 1-3.

6. Voltage and current in reduced circuits

recover branch currents and voltages after finding the total equivalent resistance,
introduce voltage-divider and current-divider reasoning,
connect this to tasks asking for current through a chosen resistor or ammeter.

7. Kirchhoff’s current law

formulate node balance using charge conservation,
analyze currents at junctions,
use KCL when simple reduction is not sufficient.

8. Kirchhoff’s voltage law

formulate loop balance using energy conservation,
track voltage rises and drops consistently around a loop,
show how KVL complements KCL in practical circuit solving.

9. Multi-branch circuit solving strategy

choose unknown currents or node potentials,
write a minimal set of independent equations,
solve for currents, resistor voltages, and point-to-point potential differences.

10. Bridge and nontrivial resistor networks

discuss circuits that are not reducible by naive series-parallel rules,
explain symmetry arguments when they apply,
prepare students for point-to-point voltage questions like the A-B problem.

11. Ammeters, voltmeters, and measurement interpretation

explain what an ideal ammeter and ideal voltmeter measure,
relate instrument placement to the quantity being measured,
interpret several ammeter-reading problems from the repository.

12. Worked examples and problem-solving workflow

one example focused on equivalent resistance,
one example focused on branch currents and resistor voltages,
one example focused on Kirchhoff equations in a nontrivial network,
final checklist: simplify if possible, label directions, write equations, solve, then interpret signs.

--- title: Circuits format: html: theme: flatly toc: true toc-depth: 3 highlight-style: tango code-line-numbers: true code-fold: true code-summary: "Show the code" code-tools: true code-block-bg: "rgba(42, 174, 42, 0.02)" code-block-border-left: "#2aae2a" code-language-label: true css: styles.css math: mathjax self-contained: true other-links: - text: Main page href: https://dchorazkiewicz.github.io/Mathematics_Physics_Lectures --- ## Scope of this lecture This lecture introduces direct-current circuit analysis from the simplest resistor combinations to multi-loop networks that require systematic equations. The plan follows the progression suggested by the problem set: first identify structure, then reduce circuits when possible, and finally compute unknown currents and voltages in circuits that can no longer be simplified by inspection alone. Main themes: - electric current, voltage, resistance, and power, - Ohm's law as a local relation for a circuit element, - series and parallel resistor combinations, - equivalent resistance as a model simplification tool, - current and voltage division in simple networks, - Kirchhoff's current law and Kirchhoff's voltage law, - systematic analysis of junctions, loops, and bridge-like circuits, - interpretation of ammeter and voltmeter readings, - choosing an efficient solution strategy before calculating. ## Lecture plan ### 1. Basic circuit quantities and conventions - define charge flow, current, potential difference, resistance, and power, - establish sign conventions for current direction and voltage drop, - explain ideal wires, ideal sources, and ideal measuring instruments. ### 2. Ohm's law and the meaning of resistance - present $$ V = IR $$ - distinguish between element law and whole-circuit reasoning, - show how resistance controls current for a given applied voltage. ### 3. Series connections - identify when elements carry the same current, - derive the equivalent resistance rule for series combinations, - connect the rule to the first problems on direct resistor reduction. ### 4. Parallel connections - identify when elements share the same voltage, - derive the equivalent resistance rule for parallel combinations, - compare parallel behavior with series behavior physically and mathematically. ### 5. Mixed series-parallel networks - develop a reduction workflow for circuits that can be simplified step by step, - emphasize redrawing the circuit after each reduction, - prepare students for equivalent-resistance tasks like Problems 1-3. ### 6. Voltage and current in reduced circuits - recover branch currents and voltages after finding the total equivalent resistance, - introduce voltage-divider and current-divider reasoning, - connect this to tasks asking for current through a chosen resistor or ammeter. ### 7. Kirchhoff's current law - formulate node balance using charge conservation, - analyze currents at junctions, - use KCL when simple reduction is not sufficient. ### 8. Kirchhoff's voltage law - formulate loop balance using energy conservation, - track voltage rises and drops consistently around a loop, - show how KVL complements KCL in practical circuit solving. ### 9. Multi-branch circuit solving strategy - choose unknown currents or node potentials, - write a minimal set of independent equations, - solve for currents, resistor voltages, and point-to-point potential differences. ### 10. Bridge and nontrivial resistor networks - discuss circuits that are not reducible by naive series-parallel rules, - explain symmetry arguments when they apply, - prepare students for point-to-point voltage questions like the A-B problem. ### 11. Ammeters, voltmeters, and measurement interpretation - explain what an ideal ammeter and ideal voltmeter measure, - relate instrument placement to the quantity being measured, - interpret several ammeter-reading problems from the repository. ### 12. Worked examples and problem-solving workflow - one example focused on equivalent resistance, - one example focused on branch currents and resistor voltages, - one example focused on Kirchhoff equations in a nontrivial network, - final checklist: simplify if possible, label directions, write equations, solve, then interpret signs.