---
title: "AP Physics C: E&M 11.3: Resistance, Resistivity, and Ohm's Law"
description: "Review AP Physics C: E&M Topic 11.3, including resistance, resistivity, Ohm's law, R = ρℓ/A, I = ΔV/R, I-V graphs, ohmic materials, temperature effects, and experimental resistance analysis."
canonical: "https://fiveable.me/ap-physics-c-e-m/unit-11/3-resistance-resistivity-and-ohms-law/study-guide/TnRPkql9C75GQe0d"
type: "study-guide"
subject: "AP Physics C: E&M"
unit: "Unit 11 – Electric Circuits"
lastUpdated: "2026-06-09"
---

# AP Physics C: E&M 11.3: Resistance, Resistivity, and Ohm's Law

## Summary

Review AP Physics C: E&M Topic 11.3, including resistance, resistivity, Ohm's law, R = ρℓ/A, I = ΔV/R, I-V graphs, ohmic materials, temperature effects, and experimental resistance analysis.

## Guide

[Resistance](/ap-physics-c-e-m/unit-11/4-electric-power/study-guide/u2cRqQTlthIAJtwp "fv-autolink") measures how strongly an object opposes the flow of electric charge, and it depends on the material's resistivity, the object's length, and its cross sectional area through $R = \frac{\rho \ell}{A}$. Ohm's law, $I = \frac{\Delta V}{R}$, connects current, [potential difference](/ap-physics-c-e-m/key-terms/potential-difference "fv-autolink"), and resistance for circuit elements, and ohmic materials keep a constant resistance shown by a straight line through the origin on an I-V graph.

## Why This Matters for the AP Physics C: E&M Exam

This topic gives you the tools to predict how current responds to potential difference and to connect a [resistor](/ap-physics-c-e-m/key-terms/resistor "fv-autolink")'s physical shape to its electrical behavior. Both the resistance formula and Ohm's law show up across the circuit unit, which carries a large share of the multiple-choice section, so being fast with them helps everywhere from simple circuits to compound circuits and RC circuits.

The free-response section includes an experimental design and analysis question. The data-analysis skills here, plotting current versus potential difference, finding resistance from a slope, and linearizing data, are exactly the kind of reasoning that question rewards. You may be asked to design a procedure to test whether a material is ohmic, collect and graph data, and justify a claim from your graph.

## Key Takeaways

- Resistance comes from material and shape: $R = \frac{\rho \ell}{A}$. Longer means more resistance, wider cross-section means less.
- Resistivity $\rho$ is a material property measured in ohm-meters, and for conductors it usually increases as temperature rises.
- Ohm's law is $I = \frac{\Delta V}{R}$. Ohmic materials have constant resistance and a linear I vs V graph through the origin.
- On an I vs V graph, the slope equals $\frac{1}{R}$, so a steeper line means lower resistance. On a V vs I graph, the slope equals $R$.
- When resistivity varies along the length, integrate: $R = \int \frac{\rho(\ell)\, d\ell}{A}$.
- Resistors convert electrical energy to thermal energy, and this heating can change the temperature of the resistor and its surroundings.

## Resistance from Physical Properties

For an object with uniform geometry, resistance depends on both the material and the object's dimensions:

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

Resistance increases when the resistivity $\rho$ increases, increases when the length $\ell$ increases, and decreases when the cross-sectional area $A$ increases. A longer path gives charges more material to move through, while a larger area provides more room for charge to flow.

- Resistivity ($\rho$) is a fundamental property of a material that depends on its atomic and molecular structure and quantifies how strongly the material opposes the motion of electric charge.
  - Resistivity is measured in ohm-meters (Ω·m).
  - For conductors, resistivity typically increases as temperature increases, so the resistance of a metal wire usually increases when it gets hotter.
- If the resistor has uniform cross-sectional area $A$ but the resistivity varies along its length, the total resistance is found by summing small pieces:

$$R = \int \frac{\rho(\ell)\, d\ell}{A}$$

## Electrical Characteristics of Circuit Elements

### Ohm's Law in Circuits

Ohm's law relates current, resistance, and potential difference across a conductive circuit element. The current through the element is directly proportional to the potential difference across it, with resistance as the proportionality constant.

- Ohm's law is written as $$I=\frac{\Delta V}{R}$$, where:
  - $$\Delta V$$ is the potential difference ([voltage](/ap-physics-c-e-m/key-terms/voltage "fv-autolink")) measured in volts (V)
  - $$I$$ is the current measured in amperes (A)
  - $$R$$ is the resistance measured in ohms (Ω)
- This is equivalent to the familiar form $$\Delta V = IR$$.
- Ohmic materials maintain a constant resistance regardless of the current passing through them.
  - For an ohmic material, current is proportional to potential difference, so the material has constant resistance and a constant resistivity. In the model used here, the resistivity of an ohmic material is treated as constant regardless of temperature.
  - Their current-voltage ($$I$$-$$V$$) graph is a straight line passing through the origin.
  - Examples include most metals over limited temperature ranges.
- Non-ohmic materials have resistance that varies with current or voltage.
  - Their $$I$$-$$V$$ graph is non-linear.
  - Examples include diodes, transistors, and light bulbs when their temperature changes.
- The resistance of an ohmic circuit element can be determined from the slope of a graph of current as a function of potential difference:
  - On a graph of current $$I$$ as a function of potential difference $$\Delta V$$ for an ohmic element, the slope is $$\frac{\Delta I}{\Delta V} = \frac{1}{R}$$. The resistance is the reciprocal of the slope: $$R = \frac{1}{(\Delta I / \Delta V)}$$.
  - A steeper slope on an $$I$$ vs. $$\Delta V$$ graph means **lower** resistance.
  - If voltage is graphed as a function of current (a $$V$$ vs. $$I$$ graph), the slope is $$\frac{\Delta V}{\Delta I} = R$$.

### Experimental Determination of Resistance

To determine whether a circuit element is ohmic and to find its resistance, vary the potential difference across the element, measure the current for several trials, and plot current $$I$$ (in amperes) versus potential difference $$\Delta V$$ (in volts).

- If the graph is linear and passes through the origin, the element is ohmic under those conditions.
- The slope of an $$I$$ versus $$\Delta V$$ graph is $$\frac{1}{R}$$, so the resistance is the reciprocal of the slope.
- To make the conclusion reliable, keep physical conditions such as temperature as constant as possible while collecting the data.
- Resistors convert electrical energy to thermal energy.
  - This power follows $$P = I^2R = \frac{\Delta V^2}{R} = I\Delta V$$.
  - This heating can change the temperature of the resistor and its surroundings.
  - For example, a light bulb filament heats up and glows when current flows through it.

## How to Use This on the AP Physics C: E&M Exam

### Problem Solving

- Pick the right form of the equation for what you are given. Use $R = \frac{\rho \ell}{A}$ when you have material and geometry, and use $I = \frac{\Delta V}{R}$ when you have circuit quantities.
- For "factor of change" questions, write a ratio. If length doubles with the same material and area, $R$ doubles, so for fixed $\Delta V$ the current halves.
- Carry units through every step. Resistivity in ohm-meters, length in meters, and area in square meters give resistance in ohms.
- When resistivity is given as a function of position, set up the integral $R = \int \frac{\rho(\ell)\, d\ell}{A}$ instead of plugging into the simple formula.

### Free Response

- For the experimental design and analysis question, describe a clear procedure: vary $\Delta V$, measure $I$ at several settings, and plot the data.
- Justify whether a material is ohmic using your graph. A straight line through the origin supports an ohmic claim; curvature supports a non-ohmic claim.
- Find resistance from a slope and state it correctly. On an I vs V graph the slope is $\frac{1}{R}$, so take the reciprocal.
- Mention controlling temperature as a way to reduce error, since heating changes resistivity for many materials.

### Common Trap

- Reading resistance directly as the slope of an I vs V graph. The slope is $\frac{1}{R}$, not $R$.

## Practice Problem 1: Ohm's Law Application

> A 12V battery is connected to a circuit containing a resistor. If the current flowing through the circuit is 2A, what is the resistance of the resistor? If the length of the resistor is doubled while keeping the same material and cross-sectional area, what happens to the resistance and the current?

**Solution**

To find the resistance, apply Ohm's law: $$\Delta V = IR$$

Rearranging for resistance: $$R = \frac{\Delta V}{I} = \frac{12\text{ V}}{2\text{ A}} = 6\text{ Ω}$$

When the length is doubled while keeping the same material and cross-sectional area, use $$R = \frac{\rho \ell}{A}$$:
- The original resistance is $$R_1 = \frac{\rho \ell_1}{A}$$
- The new resistance is $$R_2 = \frac{\rho (2\ell_1)}{A} = 2 \times \frac{\rho \ell_1}{A} = 2R_1$$

So the resistance doubles to 12 Ω.

Using Ohm's law again with the new resistance:
$$I_2 = \frac{\Delta V}{R_2} = \frac{12\text{ V}}{12\text{ Ω}} = 1\text{ A}$$

When the length doubles, the resistance doubles and the current is reduced by half.

## Practice Problem 2: I-V Graph Analysis

> A student measures the current through a circuit element at different voltages and plots the following data points: (2V, 0.4A), (4V, 0.8A), (6V, 1.2A), (8V, 1.6A). Is this an ohmic material? What is its resistance?

**Solution**

To determine if this is an ohmic material, check whether the relationship between potential difference and current is linear.

Calculate the resistance at each data point using $$R = \frac{\Delta V}{I}$$:
- At (2V, 0.4A): $$R = \frac{2\text{ V}}{0.4\text{ A}} = 5\text{ Ω}$$
- At (4V, 0.8A): $$R = \frac{4\text{ V}}{0.8\text{ A}} = 5\text{ Ω}$$
- At (6V, 1.2A): $$R = \frac{6\text{ V}}{1.2\text{ A}} = 5\text{ Ω}$$
- At (8V, 1.6A): $$R = \frac{8\text{ V}}{1.6\text{ A}} = 5\text{ Ω}$$

Since the resistance is constant (5 Ω) at all values, and the I-V relationship is linear (current doubles when potential difference doubles), this is an ohmic material.

The resistance is 5 Ω. On a graph of $$I$$ vs. $$\Delta V$$, the slope is $$\frac{\Delta I}{\Delta V} = \frac{1.6\text{ A} - 0.4\text{ A}}{8\text{ V} - 2\text{ V}} = \frac{1.2\text{ A}}{6\text{ V}} = 0.2\text{ A/V}$$. Since the slope equals $$\frac{1}{R}$$, the resistance is $$R = \frac{1}{0.2\text{ A/V}} = 5\text{ Ω}$$.

## Common Misconceptions

- "Resistance and resistivity are the same thing." Resistivity $\rho$ is a property of the material itself, while resistance $R$ also depends on the shape and size of the object through $R = \frac{\rho \ell}{A}$.
- "The slope of an I vs V graph is the resistance." The slope is $\frac{1}{R}$. You have to take the reciprocal to get resistance.
- "Every material obeys Ohm's law." Only ohmic materials have constant resistance. Diodes, transistors, and heated filaments are non-ohmic and have curved I-V graphs.
- "Resistivity never changes." For most conductors, resistivity increases as temperature rises. The constant-resistivity idea applies to the idealized ohmic model.
- "A wider resistor has more resistance because there is more material." Increasing the cross-sectional area lowers resistance because there is more room for charge to flow.
- "Ohm's law lets current flow without any energy cost." Resistors convert electrical energy to thermal energy, and that heating can raise the temperature of the resistor and its surroundings.

## Related AP Physics C: E&M Guides

- [11.1 Electric Current](/ap-physics-c-e-m/unit-11/1-electric-current/study-guide/9YRMrkv1PVy23BzH)
- [11.2 Electric Circuits](/ap-physics-c-e-m/unit-11/2-electric-circuits/study-guide/17WyJIXaesWwOEX8)
- [11.5 Compound Direct Current Circuits](/ap-physics-c-e-m/unit-11/5-compound-direct-current-circuits/study-guide/lvbJLaPd4EqBAUf6)
- [11.8 Resistor-Capacitor (RC) Circuits](/ap-physics-c-e-m/unit-11/8-resistor-capacitor-rc-circuits/study-guide/qy6QreLu93jx043L)
- [11.4 Electric Power](/ap-physics-c-e-m/unit-11/4-electric-power/study-guide/u2cRqQTlthIAJtwp)
- [11.6 Kirchhoff's Loop Rule](/ap-physics-c-e-m/unit-11/6-kirchhoffs-loop-rule/study-guide/Nxj12AcsMRzP3ejn)

## Vocabulary

- **Ohm's law**: A fundamental relationship stating that current through a conductor is directly proportional to the potential difference across it and inversely proportional to its resistance, expressed as I = ΔV/R.
- **conductor**: A material that allows electric charge to move through it, with resistivity that typically increases with temperature.
- **cross-sectional area**: The area of the surface perpendicular to the direction of current flow through a conductor.
- **current**: The flow of electric charge through a conductor, measured as the amount of charge passing through a cross-section per unit time.
- **electric potential difference**: The difference in electric potential energy per unit charge between two points in a circuit, measured in volts.
- **ohmic materials**: Materials that obey Ohm's law and maintain constant resistance regardless of the current flowing through them.
- **resistance**: The opposition to the flow of electric current in a circuit, measured in ohms (Ω).
- **resistivity**: A fundamental property of a material that quantifies how strongly the material opposes the motion of electric charge, depending on the material's atomic and molecular structure.
- **resistor**: A circuit element that dissipates electrical energy and opposes the flow of current, characterized by resistance R.
- **thermal energy**: Energy dissipated in the form of heat when electrical energy is converted within a circuit element.
- **uniform geometry**: A resistor with constant cross-sectional area and composition throughout its length.

## FAQs

### What is resistance in AP Physics C: E&M?

Resistance measures how strongly an object opposes electric charge flow. It depends on material resistivity, length, and cross-sectional area.

### What is the formula for resistance and resistivity?

For uniform geometry, resistance is R = rho l / A. Resistance increases with resistivity and length, and decreases as cross-sectional area increases.

### What does Ohm's law say?

Ohm's law relates current, potential difference, and resistance: I = Delta V / R, or equivalently Delta V = IR.

### How do you identify an ohmic material from a graph?

An ohmic material has a linear current-versus-potential-difference graph through the origin. On an I vs V graph, slope equals 1/R.

### How does temperature affect resistivity?

For most conductors, resistivity increases as temperature increases. In the ideal ohmic model for this topic, resistivity is treated as constant.

### When do you integrate to find resistance?

Use R = integral rho(l) dl / A when the object has uniform cross-sectional area but resistivity varies along its length.

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