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What determines a conductor's resistance, and when does current grow in step with voltage?

Topic 11.3 Resistance, Resistivity, and Ohm's Law: apply Ohm's law and relate resistance to resistivity, length and cross-sectional area.

A focused answer to AP Physics 2 Topic 11.3, covering resistance and Ohm's law V = IR, the dependence of resistance on resistivity, length and cross-sectional area, the meaning of ohmic and non-ohmic behavior, and how to read a current-voltage graph, with full worked examples.

Generated by Claude Opus 4.811 min answerUpdated 2026-06-04

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Quick answer

Resistance $R$ measures how strongly a component opposes current; it is the ratio of voltage to current, $R = \dfrac{V}{I}$ , measured in ohms. Ohm's law, $V = IR$ , says that for an ohmic device the resistance is constant, so current is proportional to voltage and the current-voltage graph is a straight line through the origin with slope $1/R$ . The resistance of a wire depends on the material and shape: $R = \dfrac{\rho L}{A}$ , where $\rho$ is the resistivity (a material property), $L$ the length and $A$ the cross-sectional area. So a longer or thinner wire has more resistance, and a thicker or shorter one has less. A non-ohmic device (like a filament bulb or diode) has a curved $I$ - $V$ graph because its resistance changes with conditions. Resistance is what converts a voltage into a definite current and dissipates electrical energy as heat.

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What this topic is asking
Resistance and Ohm's law
What sets a wire's resistance
Ohmic versus non-ohmic, and the I-V graph
Try this

What this topic is asking

The College Board (Topic 11.3) wants you to apply Ohm's law, $V = IR$ , relate resistance to resistivity, length and cross-sectional area ( $R = \dfrac{\rho L}{A}$ ), and distinguish ohmic from non-ohmic behavior using a current-voltage graph.

Resistance and Ohm's law

Resistance is the circuit's "friction": it sets how much current a given voltage drives. Ohm's law, $V = IR$ , is the most-used relation in the unit, rearranged as needed to find any one of voltage, current or resistance. The key qualification is "ohmic": Ohm's law holds when $R$ stays constant, which is true for many resistors at fixed temperature but not for all devices.

What sets a wire's resistance

The geometry makes physical sense: a longer wire is a longer obstacle course for the charges, so more resistance; a thicker wire gives the charges more room (more parallel paths), so less resistance. Resistivity $\rho$ is the intrinsic property of the material, separating "what it is made of" from "what shape it is." This formula is why power lines are thick (low resistance) and why a thin filament gets hot (high resistance concentrates the energy).

Ohmic versus non-ohmic, and the I-V graph

The current-voltage ( $I$ - $V$ ) graph reveals the behavior. An ohmic device gives a straight line through the origin: current proportional to voltage, constant resistance, slope $1/R$ . A non-ohmic device gives a curved graph because its resistance changes, for example a filament bulb whose resistance rises as it heats, or a diode that conducts in only one direction. Reading the graph is an exam skill: the resistance at any point is $V/I$ (the value, not the slope, unless the line passes through the origin). The strategic role of this topic is that resistance and Ohm's law are the engine of circuit calculation: combined with the current of Topic 11.1 and the emf of Topic 11.2, $V = IR$ lets you find the current anywhere, the series and parallel combinations of Topic 11.5 reduce complex networks, and the power $P = IV$ of Topic 11.4 follows directly. Almost every circuit answer comes back to $V = IR$ .

Resistance of a heating wire

A nichrome wire ( $\rho = 1.1 \times 10^{-6}$ ohm meters) is $1.5$ m long with cross-sectional area $2.0 \times 10^{-7}$ m squared. Find its resistance, then the current when $6.0$ V is applied.

step 1 Resistance from geometry

$R = \dfrac{\rho L}{A} = \dfrac{(1.1 \times 10^{-6})(1.5)}{2.0 \times 10^{-7}} = \dfrac{1.65 \times 10^{-6}}{2.0 \times 10^{-7}} = 8.25$ ohms.

step 2 Current from Ohm's law

$I = \dfrac{V}{R} = \dfrac{6.0}{8.25} = 0.73$ A.

step 3 Effect of doubling the length

A wire twice as long would have $R = 16.5$ ohms (resistance proportional to length), halving the current to $0.36$ A.

step 4 Interpret

Nichrome's high resistivity gives a usefully large resistance in a short wire, which is why it is used in heating elements: the resistance dissipates energy as heat.

Try this

Q1. A resistor has $9.0$ V across it and carries $3.0$ A. Calculate its resistance. [2 points]

Cue. $R = V/I = 9.0/3.0 = 3.0$ ohms.

Q2. State what happens to a wire's resistance if its length is doubled (same material and thickness). [1 point]

Cue. It doubles (resistance is proportional to length).

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2024 (style)6 marksSection II (short FRQ). A wire of resistivity

1.7 \times 10^{-8}

ohm meters has length

2.0

m and cross-sectional area

5.0 \times 10^{-7}

m squared. (a) Calculate its resistance. (b) A potential difference of

1.5

V is applied across it. Calculate the current. (c) State and justify what happens to the resistance if the wire is replaced by one of the same material and length but twice the cross-sectional area.

Show worked answer →

A 6-point FRQ on resistance and Ohm's law.

(a) Resistance (2 points): $R = \dfrac{\rho L}{A} = \dfrac{(1.7 \times 10^{-8})(2.0)}{5.0 \times 10^{-7}} = \dfrac{3.4 \times 10^{-8}}{5.0 \times 10^{-7}} = 0.068$ ohms.
(b) Current (2 points): $I = \dfrac{V}{R} = \dfrac{1.5}{0.068} = 22$ A.
(c) Doubling area (2 points): resistance is inversely proportional to area, $R = \rho L / A$ , so doubling the area halves the resistance to $0.034$ ohms (a thicker wire conducts more easily).

Markers reward the resistivity formula, Ohm's law, and the inverse dependence on area.

AP 2023 (style)1 marksSection I (multiple choice). A device obeys Ohm's law. Its current-voltage graph is best described as (A) a curve that bends upward (B) a straight line through the origin (C) a horizontal line (D) a vertical line. Justify your reasoning.

Show worked answer →

A 1-point MCQ on ohmic behavior. The answer is (B).

An ohmic device has $V = IR$ with constant $R$ , so current is proportional to voltage: the $I$ - $V$ graph is a straight line through the origin whose slope is $1/R$ . The trap is (A): a curved graph indicates non-ohmic behavior, where resistance changes with voltage.

Related dot points

Sources & how we know this

AP Physics 2: Algebra-Based Course and Exam Description — College Board (2024)