Resistance, Resistivity, and Ohm's Law - AP Physics 2: Algebra-Based Study Guide

Getting Started

We will investigate the phenomenon of electrical "friction" that occurs when charges move through a material. Our system is a conductive object, like a simple wire or a specially designed resistor, through which an electric current flows. We will explore the factors, from the atomic scale of a material to the macroscopic shape of an object, that determine how much it impedes charge flow and how this opposition governs the relationship between voltage and current in a circuit.

What You Should Be Able to Do

After studying this section, you should be able to:

• Calculate the resistance of a wire or bar of uniform shape given its dimensions and the material it is made from.

• Predict how changing a resistor's length, cross-sectional area, or material will affect its resistance.

• Use Ohm's law to calculate the current, potential difference, or resistance for a component in a simple circuit.

• Distinguish between ohmic and non-ohmic materials by analyzing a graph of current versus potential difference.

Key Concepts & Mechanisms

System & Preconditions

Our system is a conductive element with a potential difference applied across its ends. To simplify our model, we make several idealizations:

• The object has a uniform geometric shape, such as a cylinder or rectangular bar with a constant cross-sectional area.

• The material is homogeneous, meaning its properties are the same throughout.

• The temperature of the object remains constant.

• The flow of charge is steady, constituting a direct current (DC).

Key Steps / Relations

1. Microscopic Interaction and Opposition: An applied potential difference creates an electric field within the conductor, exerting a force on free charges (typically electrons) and causing them to accelerate. However, these moving charges do not travel unimpeded. They frequently collide with the relatively fixed atoms and ions that form the crystal lattice of the material. Each collision interrupts the electron's motion, effectively creating an opposition or "drag" against the overall flow of charge.

2. Quantifying Material Opposition (Resistivity): This intrinsic opposition to charge flow is a fundamental property of a material called resistivity, symbolized by the Greek letter rho ($\rho$). Its SI unit is the ohm-meter ($\Omega \cdot$ m). Materials with a low resistivity, like copper or silver, are excellent conductors because their atomic structure allows charge to move with relatively few collisions. Materials with a high resistivity, like glass or rubber, are insulators because their structure strongly impedes the movement of charge.

3. From Material Property to Object Property (Resistance): The total opposition of a specific object, called its resistance ($R$), depends not only on its material (resistivity) but also on its physical shape.

   • Length ($l$): A longer object provides a longer path for charges to travel, leading to more collisions. Therefore, resistance is directly proportional to length ($R\propto l$).

   • Cross-Sectional Area ($A$): A wider object provides more available paths for the charge to flow simultaneously, much like a wider pipe allows more water to flow. Therefore, resistance is inversely proportional to the cross-sectional area ($R\propto 1/A$).

   • Combining these factors gives the equation for the resistance of an object with uniform geometry:

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

     where $R$ is the resistance in ohms ($\Omega$), $\rho$ is the resistivity in $\Omega \cdot$ m, $l$ is the length in meters (m), and $A$ is the cross-sectional area in square meters (m²).

4. Circuit-Level Cause and Effect (Ohm's Law): The relationship between the "push" on the charges and the resulting flow is described by Ohm's law. The potential difference ($\Delta V$), measured in volts (V), is the cause—the work available per unit charge to move it through the object. The resistance ($R$) is the opposition. The resulting electric current ($I$), or rate of charge flow in amperes (A), is the effect. This causal relationship is expressed as:

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

   This equation shows that for a given potential difference, a higher resistance will result in a lower current.

Outputs & Effects

• The primary output is a stable electric current ($I$), the net flow of charge through the object.

• A critical side effect of the microscopic collisions is the transfer of energy from the moving charges to the material's lattice, increasing its thermal energy. This is why resistors get warm during operation.

• As charge moves through a resistor, electrical potential energy is converted to thermal energy, resulting in a drop in electric potential across the resistor in the direction of the current.

Regulation & Limits

• Ohm's law is an empirical model, not a fundamental law of nature. It holds true for materials where the resistance $R$ is constant over a wide range of applied potential differences. These are called ohmic materials.

• Many important components are non-ohmic, meaning their resistance changes as the voltage or current changes. A common example is the filament of an incandescent light bulb, whose resistance increases significantly as it heats up.

• The resistivity ($\rho$) of most materials is temperature-dependent. Our equations assume a constant operating temperature. For precise applications, this temperature dependence must be considered.

Key Models & Diagrams

The connection from a material's physical properties to its behavior in a circuit can be modeled as a chain of dependencies.

Key Components & Evidence

• Potential Difference ($\Delta V$): The work per unit charge that drives the flow; the "electrical pressure." Measured in volts (V).

• Electric Current ($I$): The rate of flow of electric charge ($I=\Delta q/\Delta t$). Measured in amperes (A).

• Resistance ($R$): A measure of an object's opposition to the flow of electric current. Measured in ohms ($\Omega$).

• Resistivity ($\rho$): An intrinsic property of a material that quantifies its opposition to charge flow. Measured in ohm-meters ($\Omega \cdot$ m).

• Length ($l$): The dimension of the resistor parallel to the direction of current flow. Measured in meters (m).

• Cross-Sectional Area ($A$): The area of the resistor's face perpendicular to the direction of current flow. Measured in square meters (m²).

• Ohm's Law ($I=\Delta V/R$): An empirical model that relates current, voltage, and resistance for ohmic materials.

• Ohmic Material: A material for which the ratio of potential difference to current (the resistance) is constant.

• Lab Evidence: Applying a variable voltage source to a resistor and measuring the resulting current produces a graph of $I$ vs. $\Delta V$. If the graph is a straight line passing through the origin, the resistor is ohmic, and its resistance is the reciprocal of the slope ($R=1/\text{slope}$).

Skill Snapshots

• Causation:

  • An applied potential difference causes an electric field within a conductor, which in turn causes a net movement of charge (current).

  • Collisions between charge carriers and the atomic lattice of a material cause an opposition to the current, which we define as resistance.

  • Increasing the length of a conductive wire causes its total resistance to increase because the path for potential collisions becomes longer.

• Comparison:

  • A conductor like aluminum has a very low resistivity, whereas a semiconductor like silicon has a moderate resistivity, and an insulator like quartz has an extremely high resistivity.

  • An ohmic resistor exhibits a constant resistance, resulting in a linear I-V graph, while a non-ohmic diode has a resistance that changes dramatically with voltage, resulting in a curved I-V graph.

  • A 10 cm wire with a 2 mm² cross-section has less resistance than a 20 cm wire with a 1 mm² cross-section made of the same material.

• Change Over Time (in an experiment):

  • Baseline: With zero potential difference across a resistor, there is no net current ($I=0$).

  • Change 1: As the potential difference across an ohmic resistor is steadily increased, the current passing through it increases in direct proportion.

  • Change 2: If the temperature of the resistor increases significantly (e.g., a light bulb filament), its resistance will typically increase, causing the current to be lower than Ohm's law would predict based on its cold resistance.

  • Continuity: For an ideal ohmic resistor held at a constant temperature, its resistance $R$ is a constant value that does not change as voltage and current vary.

Common Misconceptions & Clarifications

1. Misconception: Resistance is caused by electrons colliding with each other.

   • Clarification: While electron-electron collisions occur, the primary source of resistance in most conductors is the collision of moving electrons with the much more massive and relatively stationary positive ions of the material's crystal lattice.

2. Misconception: Batteries are sources of constant current.

   • Clarification: An ideal battery is a source of constant potential difference (voltage). The amount of current it supplies is not fixed; it is determined by the total resistance of the circuit connected to it ($I=\Delta V/R$).

3. Misconception: Any device that has resistance obeys Ohm's law.

   • Clarification: All materials and components have some resistance (except superconductors). However, Ohm's law specifically describes the special case where this resistance is constant. Many devices, such as diodes, transistors, and light bulbs, are non-ohmic because their resistance changes with the current or voltage.

4. Misconception: Resistivity and resistance are interchangeable terms.

   • Clarification: Resistivity ($\rho$) is an intrinsic property of a material—it tells you how resistive that substance is in general. Resistance ($R$) is an extrinsic property of a specific object that depends on both its material ($\rho$) and its geometry (length and area). Two wires made of copper have the same resistivity, but the longer, thinner wire will have a greater resistance.

One-Paragraph Summary

Resistance is the measure of an object's opposition to the flow of electric charge, arising from microscopic collisions between charge carriers and the atomic lattice of the material. This opposition is quantified for a specific object by its resistance ($R$), which depends on the material's intrinsic resistivity ($\rho$) as well as the object's length ($l$) and cross-sectional area ($A$) through the relation $R=\rho l/A$. For many common materials, known as ohmic materials, the relationship between the applied potential difference ($\Delta V$), resistance, and the resulting current ($I$) is described by the simple linear model of Ohm's law: $I=\Delta V/R$. This powerful equation allows us to predict and analyze the behavior of simple electrical circuits, though it is important to recognize it is an empirical model that assumes constant temperature and does not apply to all components.