4aCurrent, Resistance, and Ohm's Law
Explore how electric charge flows as current, how materials resist that flow, and the fundamental relationship V = IR that governs all electrical circuits.
Ohm's law is the most used equation in electrical engineering. Every electronic device — from smartphones to power grids — is designed using the relationships between voltage, current, and resistance. Understanding resistivity explains why copper is used for wiring, why nichrome is used in heaters, and why silicon is used in computer chips.
Why does a thin wire get hot when current flows through it — and why does a thick wire of the same material stay cool?
Lesson Overview
Electric current (I) is the rate of flow of electric charge: I = ΔQ/Δt. The SI unit is the ampere (A = C/s). By convention, current flows in the direction positive charges would move (opposite to electron flow). Resistance (R) is the opposition to current flow, measured in ohms (Ω). Ohm's law states that for many materials, the current is proportional to the applied voltage: V = IR. The resistance of a conductor depends on its material (resistivity ρ), length L, and cross-sectional area A: R = ρL/A. Resistivity increases with temperature for metals. Conductors have low resistivity; insulators have high resistivity; semiconductors fall in between and are the basis of modern electronics.
Key Equations
Worked Examples
A charge of 30 C flows through a wire in 5.0 s. Find the current.
A resistor has a voltage of 12 V across it and a current of 0.40 A through it. Find its resistance.
A copper wire (ρ = 1.68×10⁻⁸ Ω·m) has length 2.0 m and diameter 1.0 mm. Find its resistance.
A nichrome wire (ρ = 1.10×10⁻⁶ Ω·m) has resistance 5.0 Ω and cross-sectional area 0.50 mm². Find its length.
A tungsten filament has resistance 10 Ω at 20°C. The temperature coefficient of resistivity for tungsten is α = 4.5×10⁻³ /°C. Find its resistance at 2500°C (operating temperature of a light bulb).
Guided Problems
A 9.0 V battery is connected to a 180 Ω resistor. Find the current through the resistor.
Hint: Use Ohm's law: I = V/R.
A wire has resistance 8.0 Ω. If its length is doubled and its diameter is doubled, what is the new resistance?
Hint: R = ρL/A. Doubling L doubles R; doubling diameter quadruples A, which quarters R. Net effect: R_new = R × 2 × (1/4) = R/2.
How much charge flows through a 60 W light bulb connected to 120 V in 1 hour?
Hint: First find I = P/V (or use V = IR to find I = V/R). Then Q = IΔt.
The resistivity of a metal wire increases from 1.70×10⁻⁸ Ω·m at 20°C to 2.42×10⁻⁸ Ω·m at 100°C. Find the temperature coefficient of resistivity α.
Hint: Use ρ = ρ₀[1 + α(T−T₀)] → α = (ρ−ρ₀)/[ρ₀(T−T₀)].
Explain why the resistance of a metal increases with temperature, while the resistance of a semiconductor decreases with temperature.
Hint: In metals, higher temperature → more lattice vibrations → more electron scattering → higher resistance. In semiconductors, higher temperature → more charge carriers freed → lower resistance.
Key Vocabulary
Electric Current (I)
The rate of flow of electric charge past a cross-section: I = ΔQ/Δt. Unit: ampere (A = C/s). Conventional current flows from + to −; electrons flow from − to +.
Example: A current of 2.0 A means 2.0 coulombs of charge pass a point every second.
Resistance (R)
The opposition of a material to the flow of electric current. R = V/I. Unit: ohm (Ω = V/A). Depends on material, geometry, and temperature.
Example: A 100 Ω resistor with 5 V across it carries I = V/R = 5/100 = 0.05 A = 50 mA.
Ohm's Law
For ohmic materials, the current is proportional to the applied voltage: V = IR (or I = V/R). The resistance R is constant (independent of V and I) for ohmic materials.
Example: A resistor is ohmic if its V-I graph is a straight line through the origin. A diode is non-ohmic — its resistance depends on the applied voltage.
Resistivity (ρ)
An intrinsic property of a material that describes how strongly it opposes current flow. R = ρL/A. Unit: Ω·m. Low ρ → good conductor; high ρ → insulator.
Example: Copper: ρ = 1.68×10⁻⁸ Ω·m (excellent conductor). Glass: ρ ≈ 10¹² Ω·m (excellent insulator).
Drift Velocity
The average velocity of charge carriers (electrons) in a conductor due to an applied electric field. Very slow (~mm/s), but the electric field propagates at nearly the speed of light.
Example: In a copper wire carrying 1 A, the electron drift velocity is only about 0.1 mm/s — much slower than the signal speed.
Ohmic vs Non-Ohmic
Ohmic materials obey V = IR with constant R (linear V-I graph). Non-ohmic devices (diodes, transistors, light bulbs at varying temperatures) have resistance that changes with voltage or current.
Example: A light bulb filament is non-ohmic: its resistance increases as it heats up, so the V-I relationship is not linear.
Workbook Check — Interactive Quiz
Interactive Practice — 5 Questions
Ohm's law states that:
A 24 V battery is connected to a 6.0 Ω resistor. The current is:
If the length of a wire is doubled and its cross-sectional area is halved, its resistance:
The resistivity of a metal wire increases with temperature because:
A charge of 120 C flows through a wire in 2.0 minutes. The current is:
Independent Practice
A 1.5 V battery is connected to a 75 Ω resistor. Find the current. How much charge flows in 10 minutes?
An aluminum wire (ρ = 2.82×10⁻⁸ Ω·m) has length 5.0 m and diameter 2.0 mm. Find its resistance and the voltage needed to drive 3.0 A through it.
A resistor has 6.0 V across it and 0.30 A through it. Find its resistance. Is this consistent with Ohm's law if the V-I graph is linear?
Explain why household wiring uses thick copper wires rather than thin ones. What are the safety implications of using undersized wire?
★ A copper wire (ρ₀ = 1.68×10⁻⁸ Ω·m, α = 3.9×10⁻³/°C) has resistance 0.50 Ω at 20°C. (a) Find its resistance at 100°C. (b) If 5.0 A flows through it at 100°C, find the voltage across it. (c) Compare the power dissipated at 20°C and 100°C for the same current.
ChallengeCommon Mistakes
Applying Ohm's law V = IR to non-ohmic devices like diodes or light bulbs
Ohm's law only applies to ohmic materials where R is constant. For non-ohmic devices, R = V/I still gives the resistance at that operating point, but R changes with V and I.
Confusing current direction with electron flow direction
Conventional current flows from + to − (high to low potential). Electrons flow from − to + (opposite direction). Both descriptions are valid; just be consistent.
Forgetting to convert units when using R = ρL/A (e.g., using mm² instead of m²)
Use SI units throughout: ρ in Ω·m, L in m, A in m². Convert: 1 mm² = 10⁻⁶ m²; 1 cm² = 10⁻⁴ m².
Thinking resistance depends only on material, not geometry
R = ρL/A. Resistance depends on resistivity (material), length, AND cross-sectional area. A long thin wire has much higher resistance than a short thick one of the same material.
Math Tips
Ohm's law triangle: V = IR. Cover the quantity you want: V = IR; I = V/R; R = V/I.
R = ρL/A. To increase R: use higher ρ material, longer wire, or smaller cross-section. Doubling L doubles R; doubling diameter (quadrupling A) quarters R.
Current: I = ΔQ/Δt. Total charge: Q = IΔt. In 1 hour at 1 A: Q = 1 × 3600 = 3600 C.
Temperature effect on resistance: R = R₀[1 + α(T−T₀)]. For metals, α > 0 (R increases with T). For semiconductors, α < 0 (R decreases with T).