Physics 2 · Electric Circuits · 14 min read · Updated 2026-05-11
Resistivity and Resistance — AP Physics 2
AP Physics 2 · Electric Circuits · 14 min read
1. Core Definitions: Resistance vs Resistivity★★☆☆☆⏱ 3 min
Resistance (symbol $R$, unit ohm $\Omega$) is a measure of how much a material component opposes the flow of electric current through it, conventionally defined as the ratio of the potential difference across the component to the current passing through it. Resistivity (symbol $\rho$, unit ohm-meter $\Omega \cdot \text{m}$) is an intensive intrinsic property of a material, meaning it does not depend on the size or shape of the sample, only on the type of material and its temperature.
This topic accounts for approximately 12% of the AP Physics 2 Unit 4 exam weight, and appears regularly in both multiple-choice and free-response sections, often combined with other circuit concepts like power dissipation or equivalent resistance. AP exam questions frequently test the ability to distinguish between the two properties and relate changes in wire dimensions or temperature to changes in resistance.
2. The Resistance-Resistivity Relationship★★☆☆☆⏱ 4 min
The fundamental relationship between the resistance of a uniform sample and its resistivity depends directly on the sample's geometry. For a wire of uniform cross-sectional area $A$ and length $L$, resistance is given by:
R = \rho \frac{L}{A}
Increasing the length of the wire means current passes through more resistive material, so resistance increases linearly with $L$. Increasing the cross-sectional area gives more space for charge carriers to flow, so resistance decreases inversely with $A$. This formula only applies to uniform, isotropic materials, which is the only case tested on AP Physics 2.
3. Temperature Dependence of Resistivity★★★☆☆⏱ 4 min
Resistivity of a material depends on temperature, because temperature changes the motion of charge carriers and the atoms they collide with. For most metallic conductors, increasing temperature increases the kinetic energy of lattice atoms, leading to more frequent collisions between free electrons and the lattice, so resistivity increases. For semiconductors, increasing temperature releases more charge carriers from the lattice, which more than offsets the increased collision rate, so resistivity decreases as temperature increases.
The approximate linear relationship for small temperature changes around a reference temperature $T_0$ is:
Where $\rho_0$ is the resistivity at reference temperature $T_0$ (usually 20°C), and $\alpha$ is the temperature coefficient of resistivity. For metals, $\alpha$ is positive; for semiconductors, $\alpha$ is negative. Since resistance is proportional to resistivity, the same relationship applies directly to resistance:
R(T) = R_0 [1 + \alpha (T - T_0)]
4. Ohmic vs Non-Ohmic Materials★★★☆☆⏱ 3 min
A material or component is classified as Ohmic if it obeys Ohm's law, meaning that the resistance of the component is constant regardless of the potential difference applied across it or the current passing through it, at constant temperature. For an Ohmic material, a graph of potential difference $V$ vs current $I$ is a straight line passing through the origin, with slope equal to the constant resistance $R$.
Non-Ohmic materials do not have a constant resistance; resistance changes with applied voltage or current. The most common examples are filament light bulbs (resistance increases as current heats the filament) and diodes (high resistance in one direction, low in the other). A critical point to remember: the definition of resistance as $R = V/I$ still applies to non-Ohmic materials at any given point—resistance is just not constant for different operating points.
5. Additional AP-Style Worked Problems★★★★☆⏱ 7 min
Common Pitfalls
Why: Students forget that volume is conserved when stretching, so cross-sectional area also decreases by a factor of $k$ to keep volume constant.
Why: Students mix up the definitions of intensive (resistivity) and extensive (resistance) properties.
Why: Students mostly work with metallic conductors in examples, so they forget the opposite behavior of semiconductors.
Why: Students confuse the definition of resistance with Ohm's law (which requires constant resistance).
Why: Units of resistivity are given in $\Omega \cdot \text{m}$, so all input lengths must match this unit.