Resistance, Resistivity, and Ohm's Law

The quantitative study of electric circuits began in earnest during the early nineteenth century, when scientists first gained reliable sources of steady current from voltaic piles. Before that era, electricity was largely a curiosity confined to static charges and lightning demonstrations. The pressing engineering question—how much current flows through a given conductor when a voltage is applied?—drove a generation of experimentalists to map the relationships among voltage, current, and the physical properties of wires. Their discoveries laid the groundwork for every electrical technology that followed, from the telegraph to modern integrated circuits.

Ohm's insight was deceptively simple: for a wide class of materials, the current through a conductor is directly proportional to the voltage across it. Yet this relationship, combined with the concept of resistivity—an intrinsic property of each material—allows engineers and physicists to predict circuit behavior in virtually any configuration. The central question of this lesson is: What determines how much a conductor opposes the flow of charge, and how do we quantify that opposition?

Three interrelated concepts form the foundation of this topic. Resistance is the macroscopic quantity that characterizes how much a particular component opposes current; resistivity is the microscopic material property that determines resistance for a given geometry; and Ohm's law is the empirical relationship linking voltage, current, and resistance for ohmic materials. Understanding how these three ideas connect is essential for analyzing any DC circuit on the AP Physics 2 exam.

A voltage-versus-current (V–I) graph is the clearest way to distinguish ohmic from non-ohmic devices. For an ohmic material, V is directly proportional to I, yielding a straight line through the origin whose slope equals the resistance R. Non-ohmic devices—such as diodes, thermistors, and filament bulbs—exhibit curved V–I characteristics because their effective resistance changes with temperature, voltage, or current. The diagram below illustrates both behaviors side by side.

On the AP Physics 2 exam, you should be able to identify ohmic behavior from a V–I graph and recognize that the slope of a V-vs-I line gives the resistance. If the graph curves, the device is non-ohmic, and you can still define an instantaneous resistance at any operating point as V/I, but that value changes along the curve. For most AP problems, resistors are treated as ideal ohmic devices.

The mathematical description of resistance begins with Ohm's law and extends to the resistivity equation that links a conductor's physical dimensions to its resistance. Together these two equations let you solve virtually any AP-level problem involving resistors and conducting wires.

The resistivity equation R = ρL/A can be understood by considering the microscopic picture: charge carriers (usually electrons in metals) must traverse a longer path when L increases, encountering more collisions and therefore greater opposition to flow. Similarly, a larger cross-sectional area A provides more pathways for current, effectively placing resistive elements in parallel and thereby lowering the total resistance. The factor ρ encapsulates how strongly the material's lattice structure scatters charge carriers, making it an intrinsic property independent of geometry.

The resistance of a conductor depends on three factors: the material's resistivity, its length, and its cross-sectional area. The diagram below shows how changing each of these parameters independently affects the resistance, which is a common theme in AP qualitative reasoning questions.

For metals, resistivity increases approximately linearly with temperature because thermal vibrations of the lattice ions scatter conduction electrons more frequently. The relationship is often expressed as ρ(T) = ρ₀[1 + α(T − T₀)], where α is the temperature coefficient of resistivity and T₀ is the reference temperature (typically 20 °C). While this formula is not heavily tested on AP Physics 2, you should know qualitatively that heating a metallic conductor increases its resistance, and cooling a semiconductor generally increases its resistance as well (fewer charge carriers are thermally excited into the conduction band).

A space heater uses a nichrome wire heating element connected to a 120 V outlet. The wire has a circular cross-section with a diameter of 0.50 mm and a resistivity of ρ = 1.10 × 10⁻⁶ Ω·m. What length of wire is needed so that the heater dissipates 1440 W of power?

The magnified conductor section has diameter 0.50 mm. Its total path length, if straightened, is 1.785 m; this is not the heater housing width. At the stated reference temperature, resistivity 1.10×10⁻⁶ Ω·m and area 0.1963 mm² give 10 Ω.

Ohm's law is one of the most widely used relationships in physics and electrical engineering, but it is important to understand its domain of validity. It is not a fundamental law of nature in the same way that Maxwell's equations or conservation of energy are; rather, it is an empirical observation that holds for a specific class of materials under specific conditions. The table below contrasts when Ohm's law works well with situations where it breaks down.

The ideas of resistance and resistivity covered in AP Physics 2 serve as the foundation for more advanced treatments in university-level electromagnetism and solid-state physics. Below is a comparison of the AP-level treatment with the deeper frameworks you may encounter in subsequent courses.

While the AP Physics 2 curriculum does not require knowledge of the Drude model or band theory, understanding that resistance arises from microscopic scattering events gives you valuable physical intuition. When you encounter questions about why resistance changes with temperature or why different materials have vastly different resistivities, you can reason from the idea that anything that changes the frequency or severity of electron-lattice collisions will alter the resistance.