🎯⭐ INTERACTIVE LESSON

Current, Resistance, and Ohm's Law

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Current, Resistance, and Ohm's Law - Complete Interactive Lesson

Part 1: Electric Current

⚡ Electric Current

Part 1 of 7 — Charges in Motion

So far in electrostatics, charges have been sitting still. Now we let them move — and that moving charge is called electric current.

In this part you'll learn:

What electric current is and how it's measured
The difference between conventional current and electron flow
What drift velocity means (and why it's shockingly slow)
How to calculate current from charge and time

What Is Electric Current?

Electric current is the rate at which electric charge flows past a point in a circuit.

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

Symbol	Meaning	SI Unit
$I$	Current	Ampere (A)
$\Delta Q$	Charge	Coulomb (C)
$\Delta t$	Time	Second (s)

The Ampere

$1 \text{ A} = 1 \text{ C/s}$

One ampere means one coulomb of charge passes a point every second.

Charge Carriers

In metals, the charge carriers are free electrons (conduction electrons). Each carries charge $e = 1.6 \times 10^{-19}$ C.

In electrolytes (salt water, batteries), both positive and negative ions can carry current.

In semiconductors, both electrons and "holes" (missing electrons) carry current.

Conventional Current vs. Electron Flow

The Historical Convention

Benjamin Franklin guessed (incorrectly) that positive charges flow through wires. We still use his convention:

Conventional current flows from high potential (+) to low potential (−).

The Reality

In a metal wire, electrons actually flow from − to + (opposite to conventional current).

Why Keep the Convention?

All circuit equations work perfectly with conventional current
The math doesn't care which sign you pick — as long as you're consistent
AP Physics uses conventional current unless stated otherwise

Drift Velocity

When a voltage is applied, electrons don't race through the wire. They drift slowly, bumping into atoms along the way.

The drift velocity $v_d$ is typically $\sim^{}$ m/s — about 0.1 mm/s!

Current Concepts Quiz

Current Calculation Drill ⚡

A phone charger delivers 2.0 A for 1 hour. How many coulombs of charge are transferred? (in C)
A lightning bolt transfers 5.0 C of charge in $2.0 \times 10^{-3}$ s. What is the average current? (in A)
A copper wire (cross-section $1.0 \times 10^{-6}$ , ) carries 2.0 A. What is the drift velocity? (in m/s, use scientific notation like 1.5e-4)

Exit Quiz

Part 2: Resistance & Resistivity

🔌 Resistance & Resistivity

Part 2 of 7 — Why Charges Slow Down

Current doesn't flow freely — every material resists it to some degree. Understanding resistance and resistivity lets you predict how much current a given voltage will push through any conductor.

What Is Resistance?

Resistance measures how much a material opposes the flow of electric current.

$R = \frac{V}{I}$

Symbol	Meaning

Part 3: Ohm's Law & Power

📐 Ohm's Law & Electric Power

Part 3 of 7 — The Most Important Equation in Circuits

Ohm's Law connects voltage, current, and resistance in one elegant equation. Combined with the power formulas, you can analyze any simple circuit.

Ohm's Law

$V = IR$

This says: the voltage drop across a resistor equals the current through it times its resistance.

Three Forms

Solving for	Formula
Voltage	$V = IR$

Part 4: Temperature Dependence

🌡️ Resistivity & Temperature

Part 4 of 7 — Why Hot Wires Resist More

Resistance isn't fixed — it changes with temperature. Understanding this relationship is essential for designing circuits that work reliably and for understanding exotic phenomena like superconductivity.

Temperature Dependence of Resistivity

For most metals, resistivity increases approximately linearly with temperature:

$\rho = \rho_0 \bigl(1 + \alpha \Delta T\bigr)$

Part 5: Electric Power & Energy

💡 Electric Power & Energy

Part 5 of 7 — Paying for Electrons

Every electrical device converts energy from one form to another. Understanding power and energy lets you calculate how much energy a device uses — and how much it costs to run.

Power Formulas (Review & Extension)

Recall the three forms of the power equation:

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

Part 6: EMF & Internal Resistance

🔋 Real-World Applications

Part 6 of 7 — Batteries, Bulbs, and Safety

Real circuits aren't ideal. Batteries have internal resistance, wires have finite conductivity, and too much current can be dangerous. Let's see how the theory connects to the real world.

Batteries: EMF & Internal Resistance

A real battery isn't a perfect voltage source. It has:

EMF ( $\varepsilon$ ): the "ideal" voltage the battery would supply with no current flowing (open-circuit voltage)
Internal resistance ( $r$ ): resistance inside the battery itself

Terminal Voltage

When current $I$ flows through the battery:

$V_{terminal} = ε - I$

Part 7: Synthesis & AP Review

🎯 Synthesis & AP Review

Part 7 of 7 — Putting It All Together

You've learned about current, resistance, resistivity, Ohm's Law, power, and real-world applications. Now let's connect everything and prepare for the AP exam.

Concept Map: Current, Resistance & Ohm's Law

The Core Equations

Equation	What It Describes
$I = \Delta Q / \Delta t$	Current = charge flow rate
$I = nAv_d e$

So why does a light turn on instantly? The electric field propagates at nearly the speed of light. Every electron in the wire starts moving almost simultaneously.

Current in Terms of Drift Velocity

Symbol	Meaning
$n$	Number density of charge carriers $(electrons/m^{3})$
$A$	Cross-sectional area of the wire
$v_d$	Drift velocity
$e$	Charge per carrier ( $1.6 \times 10^{-19}$ C)

For copper: $n \approx 8.5 \times 10^{28}$ $electrons/m^{3}$

n = 8.5 \times 10^{28}

Round all answers to 3 significant figures.

$1 \;\Omega = 1 \;\text{V/A}$

Think of resistance like friction for charges. As electrons drift through a conductor, they collide with the vibrating lattice of atoms. Each collision:

Transfers kinetic energy to the lattice (→ heat)
Slows the electron down before the electric field accelerates it again

More collisions → more resistance → less current for a given voltage.

Resistivity and the Resistance Formula

Resistance depends on both the material and the geometry of the conductor:

$R = \frac{\rho L}{A}$

Symbol	Meaning	SI Unit
$\rho$	Resistivity	$\Omega \cdot \text{m}$
$L$	Length	m
$A$	Cross-sectional area	$m^{2}$

What Each Factor Does

Change	Effect on $R$	Why?
Longer wire	$R$ increases	More material for electrons to travel through
Thicker wire	$R$ decreases	More "lanes" for electrons to flow
Higher $\rho$	increases

Typical Resistivities (at 20°C)

Material	$\rho$ ( $\Omega \cdot \text{m}$ )	Type
Silver	$1.59 \times 10^{-8}$

Conductors vs. Insulators vs. Semiconductors

Conductors ( $\rho \sim 10^{-8}$ ): many free electrons, very low resistance
Semiconductors ( $\rho \sim 10^{-1}$ to ): few free carriers at room temp; resistance decreases with temperature

Resistance Concepts Quiz

Resistance Proportionality Check 🎯

Resistance Calculation Drill 🔧

Use $\rho_{\text{Cu}} = 1.68 \times 10^{-8}$ $\Omega\cdot$ m.

A copper wire is 10.0 m long with cross-sectional area $2.0 \times 10^{-6}$ $m^{2}$ . What is its resistance? (in $\Omega$ )
A copper wire has resistance 0.50 and length 5.0 m. What is its cross-sectional area? (in , scientific notation like 1.7e-7)

Ohmic vs. Non-Ohmic Materials

Ohmic materials obey $V = IR$ with constant $R$ :

Metals at constant temperature
Carbon resistors
Their $I$ - $V$ graph is a straight line through the origin

Non-ohmic materials have resistance that changes:

Light bulbs (filament heats up → $R$ increases)
Diodes (current flows in only one direction)
Semiconductors
Their $I$ - $V$ graph is curved

Reading $I$ - $V$ Curves

On an $I$ vs. $V$ graph:

Slope = $1/R$ (for ohmic materials, the slope is constant)
Steeper line = lower resistance (more current for same voltage)
Shallower line = higher resistance

Electric Power

Power is the rate at which electrical energy is converted to other forms (heat, light, motion):

$P = IV$

Combining with Ohm's Law gives three equivalent forms:

Formula	Best when you know
$P = IV$	Current and voltage
$P = I^2 R$	Current and resistance
$P = V^2 / R$	Voltage and resistance

Units

$1 \text{ W} = 1 \text{ V} \cdot \text{A} = 1 \text{ A}^2 \cdot \Omega = 1 \text{ V}^2 / \Omega$

Key Insight

In a resistor, electrical energy is converted entirely to thermal energy (heat). This is called Joule heating or resistive dissipation.

A 100 W light bulb converts 100 joules of electrical energy per second — mostly to heat, with a small fraction as visible light.

Ohm's Law Concept Check

Power Concept Check

Ohm's Law & Power Drill 🔧

A 100 $\Omega$ resistor carries 0.30 A. What is the voltage across it? (in V)
A toaster draws 10 A from a 120 V outlet. What is its resistance? (in $\Omega$ )
What power does the toaster in #2 dissipate? (in W)
A 500 $\Omega$ resistor has 25 V across it. What power does it dissipate? (in W)

Round all answers to 3 significant figures.

Symbol	Meaning
$\rho$	Resistivity at temperature $T$
$\rho_0$	Resistivity at reference temperature $T_0$ (usually 20°C)
$\alpha$	Temperature coefficient of resistivity (° $C^{-1}$ )
$\Delta T$	$T - T_0$ (temperature change)

Since $R = \rho L/A$ and the geometry changes are usually negligible:

$R = R_0 \bigl(1 + \alpha \Delta T\bigr)$

Typical Temperature Coefficients

Material	$\alpha$ (° $C^{-1}$ )
Silver	$3.8 \times 10^{-3}$
Copper	$3.9 \times 10^{-3}$
Aluminum	$3.9 \times 10^{-3}$
Tungsten	$4.5 \times 10^{-3}$
Nichrome	$0.4 \times 10^{-3}$
Carbon	$-0.5 \times 10^{-3}$
Silicon	$-75 \times 10^{-3}$

Why Metals Have Positive $\alpha$

Higher temperature → atoms vibrate more → more collisions with drifting electrons → higher resistivity.

Why Semiconductors Have Negative $\alpha$

Higher temperature → more electrons gain enough energy to become free carriers → more charge carriers → lower resistivity (despite more collisions).

Superconductors

At very low temperatures, some materials have their resistance drop to exactly zero.

Key Facts

Below a critical temperature $T_c$ , resistance = 0
Current flows indefinitely with no energy loss
Mercury: $T_c = 4.2$ K (discovered 1911)
High-temperature superconductors: $T_c \sim 90$ – $130$ K (still very cold!)
Applications: MRI magnets, particle accelerators, maglev trains

Practical Wire Sizing (AWG)

American Wire Gauge (AWG) is the standard system for wire sizes in the US:

AWG	Diameter (mm)	Typical Use
14	1.63	15 A circuits (lighting)
12	2.05	20 A circuits (outlets)
10	2.59	30 A circuits (dryers)
6	4.11	60 A circuits (ranges)

Smaller AWG number = thicker wire = lower resistance = higher current capacity.

Why does wire gauge matter? If a wire is too thin for the current it carries, $P = I^2 R$ causes excessive heating — a fire hazard!

Temperature & Resistance Quiz

Temperature-Resistance Drill 🌡️

A copper wire ( $\alpha = 3.9 \times 10^{-3}$ ° $C^{-1}$ ) has resistance 5.0 $\Omega$ at 20°C. What is its resistance at 120°C? (in $\Omega$ )
An aluminum wire ( $\alpha = 3.9 \times 10^{-3}$ ° $C^{-1}$ ) has resistance 10.0 $Ω$ at 20°C. At what temperature will its resistance be 15.0 ? (in °C)
A carbon resistor ( $\alpha = -0.5 \times 10^{-3}$ ° $C^{-1}$ ) has resistance 1000 at 20°C. What is its resistance at 220°C? (in )

Round all answers to 3 significant figures.

Power is the rate of energy use, so:

Symbol	Meaning	SI Unit
$E$	Energy	Joule (J)
$P$	Power	Watt (W)
$t$	Time	Second (s)

Your electric company doesn't bill you in joules — they use kilowatt-hours (kWh):

$1 \text{ kWh} = 1000 \text{ W} \times 3600 \text{ s} = 3.6 \times 10^6 \text{ J} = 3.6 \text{ MJ}$

A kilowatt-hour is the energy used by a 1000 W device running for 1 hour.

Typical Electricity Cost

In the US: approximately $0.12–$ 0.15 per kWh (varies by region).

$\text{Cost} = \text{Energy (kWh)} \times \text{Rate (\$/kWh)}$

Household Circuits

Standard US Household Power

Property	Value
Voltage	120 V (standard) or 240 V (large appliances)
Frequency	60 Hz (AC)
Typical circuit breaker	15 A or 20 A

Maximum Power per Circuit

For a 15 A, 120 V circuit:

$P_{\text{max}} = IV = (15)(120) = 1800 \text{ W}$

This is why you can't run too many high-power devices on one circuit!

Common Appliance Power Ratings

Appliance	Power (W)
LED bulb	10
Laptop	50–100
Hair dryer	1200–1800
Microwave	1000–1200
Space heater	1500
Electric oven	2000–5000 (240 V circuit)
Central AC	3000–5000 (240 V circuit)

Why Large Appliances Use 240 V

For the same power: $P = IV$ , doubling $V$ halves $I$ . Lower current means:

Thinner (cheaper) wires
Less $I^2R$ heating loss in the wires
Smaller circuit breakers

Power & Energy Quiz

Energy & Cost Drill 💰

Use an electricity rate of $0.12/kWh.

A 100 W light bulb runs 24 hours a day for 30 days. How many kWh does it use?
What is the monthly cost for that light bulb? (in $, round to nearest cent)
A 5000 W electric dryer runs for 45 minutes. How many kWh does it use?
You replace ten 60 W incandescent bulbs with ten 9 W LED bulbs. How many kWh do you save per day if they run 6 hours/day?
At $0.12/kWh, how much do you save per year (365 days) from that LED swap? (in $, round to nearest dollar)

Round all answers to 3 significant figures.

V_{terminal} = ε - I r

The internal resistance causes a voltage drop inside the battery, so the voltage available to the external circuit is less than the EMF.

Condition	Terminal Voltage
No current ( $I = 0$ )	$V = \varepsilon$ (full EMF)
Small current	$V \approx \varepsilon$ (small drop)
Large current	$V \ll \varepsilon$ (significant drop)
Short circuit ( $R_{\text{ext}} = 0$ )	$I = \varepsilon / r$ (maximum current, dangerous!)

Current in the Full Circuit

With external resistance $R$ :

$I = \frac{\varepsilon}{R + r}$

The total resistance is always $R + r$ , and the EMF drives the current through both.

$P_{\text{total}} = \varepsilon I = I^2R + I^2r$

$I^2R$ : useful power delivered to external circuit
$I^2r$ : wasted as heat inside the battery

Circuit Safety Devices

Fuses

A fuse contains a thin wire that melts when current exceeds its rating, breaking the circuit.

One-time use — must be replaced after blowing
Rated by current (e.g., 15 A, 20 A)
The thin wire has high resistance → heats up quickly at high current

Circuit Breakers

A circuit breaker uses a bimetallic strip or electromagnet to trip (open) the circuit when current is too high.

Resettable — just flip the switch
Standard in modern homes
Also rated by current

Ground Fault Circuit Interrupter (GFCI)

A GFCI detects when current flowing out differs from current flowing back (meaning some current is leaking through a person or water).

Trips in milliseconds
Required near water: bathrooms, kitchens, outdoors
Can detect leakage as small as 5 mA

Why These Matter

Current through body	Effect
1 mA	Barely perceptible
10 mA	Painful shock
100 mA	Can be fatal (ventricular fibrillation)
1 A	Severe burns, likely fatal

Safety devices prevent dangerous currents from flowing through the circuit — and through you.

Battery & EMF Quiz

Safety Concepts Quiz

Real-World Application Check 🎯

A battery ( $\varepsilon$ ) drives current through a circuit
Current ( $I$ ) depends on total resistance: $I = \varepsilon / (R + r)$
Resistance ( $R$ ) depends on material ( $\rho$ ), length ( $L$ ), area ( $A$ ), and temperature
Power dissipated in each element: $P = I^2R$
Energy over time: $E = Pt$

Common Mistakes to Avoid

Mistake	Correction
Forgetting internal resistance	Total $R$ = $R_{\text{ext}} + r$
Using wrong power formula	Match formula to known quantities
Confusing $\rho$ (resistivity) with $R$ (resistance)	$\rho$ is a material property; $R$ depends on geometry too
Thinking drift velocity = signal speed	Drift velocity is $\sim 10^{-4}$ m/s; signal propagates at $\sim c$
Mixing up conventional current and electron flow	Conventional: + to −. Electrons: − to +
Forgetting units on $\alpha$	$\alpha$ has units of ° $C^{-1}$ $(or K^{-1})$

Mixed Concept Check

Mixed Problem Drill 🧮

A nichrome wire ( $\rho = 1.10 \times 10^{-6}\;\Omega\cdot$ m) is 2.0 m long with diameter 0.50 mm. What is its resistance? (in $\Omega$ , round to 1 decimal)
A battery ( $\varepsilon = 12$ V, $r = 0.40\;\Omega$ ) is connected to a $5.6\;\Omega$ resistor. What current flows? (in A)
What is the terminal voltage of the battery in #2? (in V)
A 1200 W hair dryer runs on 120 V for 15 minutes. How much energy does it use? (in kJ)

Round all answers to 3 significant figures.

AP FRQ Preview

On the AP Physics 2 exam, you'll encounter free-response questions that combine multiple concepts. Here's the type of reasoning you'll need:

Example FRQ Scenario

A student has a battery of unknown EMF and internal resistance. She connects it to a variable external resistor and measures both the terminal voltage and the current for several resistance values.

Part (a): Explain how to determine $\varepsilon$ and $r$ from a graph of $V$ vs. $I$ .

Key insight: $V = \varepsilon - Ir$ is a linear equation of the form $y = b + mx$ :

y-intercept ( $I = 0$ ): $V = \varepsilon$ → gives EMF
Slope: $-r$ → gives internal resistance
x-intercept ( $V = 0$ ): → gives short-circuit current

Part (b): The student wants to maximize the power delivered to the external resistor. What value of $R$ should she use?

Key insight: $P_R = I^2 R = [\varepsilon/(R+r)]^2 R$ . Taking the derivative and setting it to zero gives (maximum power transfer theorem).

Part (c): Why is the "efficiency" (fraction of power delivered externally) only 50% at maximum power transfer?

Key insight: When $R = r$ , the current is $I = \varepsilon/2r$ . Power to load: $I^2R = \varepsilon^2/4r$ . Total power: . Efficiency = 50%.

AP Exam Tips

Show your work — write the equation, substitute, solve
Include units in every answer
Justify qualitative answers with equations
Sketch graphs when asked — label axes and key features
Check limiting cases — does your answer make sense when $R \to 0$ or $R \to \infty$ ?

Final Mastery Quiz 🏆

Insulators (

\rho > 10^{8}

): almost no free carriers, extremely high resistance

You need a copper wire with resistance exactly 1.0 $\Omega$ and diameter 1.0 mm. How long must it be? (in m, round to nearest whole number)

\varepsilon I = \varepsilon^2/2r

Current, Resistance, and Ohm's Law

Current in Terms of Drift Velocity

Physical Picture

Resistivity and the Resistance Formula

What Each Factor Does

Typical Resistivities (at 20°C)

Conductors vs. Insulators vs. Semiconductors

Ohmic vs. Non-Ohmic Materials

Reading $I$ - $V$ Curves

Electric Power

Units

Key Insight

Typical Temperature Coefficients

Why Metals Have Positive $\alpha$

Why Semiconductors Have Negative $\alpha$

Superconductors

Key Facts

Practical Wire Sizing (AWG)

Energy

The Kilowatt-Hour

Typical Electricity Cost

Household Circuits

Standard US Household Power

Maximum Power per Circuit

Common Appliance Power Ratings

Why Large Appliances Use 240 V

Key Relationships

Current in the Full Circuit

Power Budget

Circuit Safety Devices

Fuses

Circuit Breakers

Ground Fault Circuit Interrupter (GFCI)

Why These Matter

How They Connect

Common Mistakes to Avoid

AP FRQ Preview

Example FRQ Scenario

AP Exam Tips

$R$	Resistance	Ohm ( $\Omega$ )
$V$	Voltage	Volt (V)
$I$	Current	Ampere (A)

Current, Resistance, and Ohm's Law

Current, Resistance, and Ohm's Law - Complete Interactive Lesson

Part 1: Electric Current

⚡ Electric Current

What Is Electric Current?

The Ampere

Charge Carriers

Conventional Current vs. Electron Flow

The Historical Convention

The Reality

Why Keep the Convention?

Drift Velocity

Part 2: Resistance & Resistivity

🔌 Resistance & Resistivity

What Is Resistance?

Part 3: Ohm's Law & Power

📐 Ohm's Law & Electric Power

Ohm's Law

Three Forms

Part 4: Temperature Dependence

🌡️ Resistivity & Temperature

Temperature Dependence of Resistivity

Part 5: Electric Power & Energy

💡 Electric Power & Energy

Power Formulas (Review & Extension)

Part 6: EMF & Internal Resistance

🔋 Real-World Applications

Batteries: EMF & Internal Resistance

Terminal Voltage

Part 7: Synthesis & AP Review

🎯 Synthesis & AP Review

Concept Map: Current, Resistance & Ohm's Law

The Core Equations

Current in Terms of Drift Velocity

Physical Picture

Resistivity and the Resistance Formula

What Each Factor Does

Typical Resistivities (at 20°C)

Conductors vs. Insulators vs. Semiconductors

Ohmic vs. Non-Ohmic Materials

Reading III-VVV Curves

Electric Power

Units

Key Insight

Typical Temperature Coefficients

Why Metals Have Positive α\alphaα

Why Semiconductors Have Negative α\alphaα

Superconductors

Key Facts

Practical Wire Sizing (AWG)

Energy

The Kilowatt-Hour

Typical Electricity Cost

Household Circuits

Standard US Household Power

Maximum Power per Circuit

Common Appliance Power Ratings

Why Large Appliances Use 240 V

Key Relationships

Current in the Full Circuit

Power Budget

Circuit Safety Devices

Fuses

Circuit Breakers

Ground Fault Circuit Interrupter (GFCI)

Why These Matter

How They Connect

Common Mistakes to Avoid

AP FRQ Preview

Example FRQ Scenario

AP Exam Tips

Reading $I$ - $V$ Curves

Why Metals Have Positive $\alpha$

Why Semiconductors Have Negative $\alpha$