AgentSkillsCN

14 Circuits

方法论

SKILL.md
--- frontmatter
id: circuits
subject: physics
display_name: Electric Circuits
description: |
  Analyzes electric circuits using Ohm's law, series/parallel resistor combinations, and Kirchhoff's laws.
  Use when students need to solve problems involving current, voltage, resistance, power, or circuit analysis.

grade_band: 9-12
khan_tags: [physics, circuits, ohms-law, resistors, power, kirchhoffs-laws]
standards:
  - NGSS.HS-PS3-3
  - NGSS.HS-PS3-5

objectives:
  - Define current as charge flow rate (I = Q/t)
  - Apply Ohm's law (V = IR) to circuit elements
  - Calculate equivalent resistance for series and parallel circuits
  - Apply Kirchhoff's voltage law (loop rule) and current law (junction rule)
  - Calculate electrical power using P = IV = I²R = V²/R
  - Analyze simple RC circuits and time constants

prerequisites:
  - electrostatics

estimated_time_minutes: 90

validator:
  type: numeric_solver
  config:
    unit_library: physics
    default_tolerance: 0.02
    require_units: true

sources:
  - name: OpenStax College Physics
    chapter: 20-21
    url: https://openstax.org/books/college-physics/pages/20-introduction-to-electric-current-resistance-and-ohms-law
    license: CC-BY-4.0

Electric Circuits

Misconceptions

MisconceptionCorrection
"Current gets used up in a circuit"Current is conserved; it flows continuously around a closed loop. The same current enters and exits a resistor
"Batteries store charge"Batteries store chemical energy and provide a voltage (potential difference), not charge
"Voltage flows through a wire"Voltage is a potential difference between two points; current flows, voltage is measured across
"All resistors get the same current"Only true in series; in parallel, current divides inversely proportional to resistance
"Adding resistors always increases total resistance"True for series, but parallel combinations decrease total resistance

Key Concepts

Electric Current

Current is the rate of charge flow through a conductor.

Definition:

  • •I = Q/t (charge per unit time)
  • •Unit: Ampere (A) = Coulomb/second (C/s)
  • •Conventional current flows from + to - (opposite to electron flow)

Resistance

Resistance opposes the flow of current.

Ohm's Law:

  • •V = IR
  • •R = V/I
  • •Unit: Ohm (Ω) = V/A

Factors affecting resistance:

  • •R = ρL/A (resistivity × length / cross-sectional area)
  • •Temperature increases resistance in most conductors

Resistors in Series

When resistors are connected end-to-end:

  • •Same current through all: I_total = I_1 = I_2 = I_3
  • •Voltages add: V_total = V_1 + V_2 + V_3
  • •Equivalent resistance: R_eq = R_1 + R_2 + R_3

Resistors in Parallel

When resistors share the same two nodes:

  • •Same voltage across all: V_total = V_1 = V_2 = V_3
  • •Currents add: I_total = I_1 + I_2 + I_3
  • •Equivalent resistance: 1/R_eq = 1/R_1 + 1/R_2 + 1/R_3

Special case for two resistors in parallel:

  • •R_eq = (R_1 × R_2) / (R_1 + R_2)

Kirchhoff's Laws

Junction Rule (KCL - Kirchhoff's Current Law):

  • •Sum of currents entering a junction = sum of currents leaving
  • •ΣI_in = ΣI_out
  • •Based on conservation of charge

Loop Rule (KVL - Kirchhoff's Voltage Law):

  • •Sum of voltage changes around any closed loop = 0
  • •ΣV = 0 (around a loop)
  • •Based on conservation of energy

Sign conventions for loop rule:

  • •EMF: + if traversing from - to + terminal
  • •Resistor: - if traversing in direction of current (voltage drop)

Electrical Power

Power is the rate of energy transfer.

Power formulas:

  • •P = IV (general form)
  • •P = I²R (useful when current is known)
  • •P = V²/R (useful when voltage is known)
  • •Unit: Watt (W) = J/s = V·A

Energy dissipated:

  • •E = Pt (energy = power × time)
  • •Unit: Joule (J) or kilowatt-hour (kWh)

RC Circuits (Basics)

A circuit with a resistor and capacitor in series.

Charging a capacitor:

  • •Q(t) = Q_max(1 - e^(-t/τ))
  • •τ = RC (time constant)
  • •After 5τ, capacitor is ~99% charged

Discharging:

  • •Q(t) = Q_0 × e^(-t/τ)
  • •Current and voltage decrease exponentially

Equations

code
[1] I = Q/t (definition of current)
[2] V = IR (Ohm's law)
[3] R_series = R_1 + R_2 + R_3 + ... (series resistors)
[4] 1/R_parallel = 1/R_1 + 1/R_2 + 1/R_3 + ... (parallel resistors)
[5] P = IV = I²R = V²/R (electrical power)
[6] ΣI_in = ΣI_out (Kirchhoff's current law)
[7] ΣV_loop = 0 (Kirchhoff's voltage law)
[8] τ = RC (RC time constant)

Worked Examples

Example 1: Ohm's Law

Problem: A 12 V battery is connected to a 4 Ω resistor. What current flows through the resistor?

Solution:

  1. •Use Ohm's law: V = IR
  2. •Solve for I: I = V/R
  3. •I = 12/4 = 3 A

Example 2: Series Circuit

Problem: Three resistors (2 Ω, 3 Ω, and 5 Ω) are connected in series to a 20 V battery. Find the current and voltage across each resistor.

Solution:

  1. •Find equivalent resistance: R_eq = 2 + 3 + 5 = 10 Ω
  2. •Find total current: I = V/R_eq = 20/10 = 2 A
  3. •Voltage drops: V_1 = IR_1 = 2×2 = 4 V, V_2 = 2×3 = 6 V, V_3 = 2×5 = 10 V
  4. •Check: 4 + 6 + 10 = 20 V ✓

Example 3: Parallel Circuit

Problem: Two resistors (6 Ω and 3 Ω) are connected in parallel to a 12 V source. Find the total current.

Solution:

  1. •Equivalent resistance: 1/R_eq = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2
  2. •R_eq = 2 Ω
  3. •Total current: I = V/R_eq = 12/2 = 6 A

Example 4: Power Calculation

Problem: A 60 W light bulb operates at 120 V. What is its resistance and the current through it?

Solution:

  1. •Find current: P = IV → I = P/V = 60/120 = 0.5 A
  2. •Find resistance: R = V/I = 120/0.5 = 240 Ω
  3. •Or directly: P = V²/R → R = V²/P = 14400/60 = 240 Ω

Explanation Patterns

  1. •Identify circuit type - Is it series, parallel, or combination?
  2. •Simplify the circuit - Combine series and parallel sections step by step
  3. •Find total current - Use V_source / R_equivalent
  4. •Work backward - Find voltage and current in each branch
  5. •Apply Kirchhoff's laws for complex circuits
  6. •Check your answer - Verify KCL at junctions and KVL around loops

Common Problem Types

  1. •Ohm's law calculations: Direct application of V = IR
  2. •Series circuits: Same current, voltages add
  3. •Parallel circuits: Same voltage, currents add
  4. •Combined circuits: Systematic simplification
  5. •Power calculations: Using P = IV, I²R, or V²/R
  6. •Kirchhoff's laws: Multi-loop circuit analysis
  7. •RC circuits: Time constant and exponential behavior