---
title: "AP Physics C: E&M 12.4: Ampère's Law"
description: "Review AP Physics C: E&M Topic 12.4, including Ampère's law, Ampere's law formula, Amperian loops, enclosed current, long straight wires, solenoids, current density, superposition, right-hand rule, and Maxwell's addition."
canonical: "https://fiveable.me/ap-physics-c-e-m/unit-12/4-amperes-law/study-guide/RURo66Hv1aueyDWX"
type: "study-guide"
subject: "AP Physics C: E&M"
unit: "Unit 12 – Magnetic Fields & Electromagnetism"
lastUpdated: "2026-06-09"
---

# AP Physics C: E&M 12.4: Ampère's Law

## Summary

Review AP Physics C: E&M Topic 12.4, including Ampère's law, Ampere's law formula, Amperian loops, enclosed current, long straight wires, solenoids, current density, superposition, right-hand rule, and Maxwell's addition.

## Guide

[Ampere's law](/ap-physics-c-e-m/key-terms/amperes-law "fv-autolink") relates the magnetic field along a closed loop to the current passing through it: $\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{enc}$. You use it to find the magnetic field in symmetric setups like long straight wires, solenoids, and cylindrical conductors by choosing a smart [Amperian loop](/ap-physics-c-e-m/key-terms/amperian-loop "fv-autolink").

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

[Unit 12](/ap-physics-c-e-m/unit-12 "fv-autolink") carries a lot of weight on the exam, and Ampere's law is one of the cleanest tools for finding magnetic fields fast. On multiple-choice and free-response questions, you may need to set up the [line integral](/ap-physics-c-e-m/key-terms/line-integral "fv-autolink"), pick an Amperian loop that matches the symmetry of the current, calculate the enclosed current, and solve for $B$. The exam also expects you to derive symbolic expressions and apply the right-hand rule to get field direction, so understanding both the math and the physical meaning pays off.

This topic connects directly to the symmetry reasoning you built with [Gauss's law](/ap-physics-c-e-m/unit-8/6-gausss-law/study-guide/HnTBd7Mh37yvO3cx "fv-autolink") for [electric fields](/ap-physics-c-e-m/unit-8/3-electric-fields/study-guide/7Nyjo6HcMeSSkleV "fv-autolink"). When a setup has enough symmetry, Ampere's law turns a hard integral into simple algebra, which is exactly the kind of efficient problem solving the exam rewards.

## Key Takeaways

- Ampere's law in integral form is $\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{enc}$, where $I_{enc}$ is only the current passing through the loop.
- For a long straight wire, the field is $B_{wire} = \dfrac{\mu_0}{2\pi}\dfrac{I}{r}$, forming concentric circles set by the right-hand rule.
- For a long ideal [solenoid](/ap-physics-c-e-m/key-terms/solenoid "fv-autolink"), the inside field is $B_{sol} = \mu_0 n I$, uniform along the axis, with negligible field outside.
- The exam only expects quantitative Ampere's law work for symmetric cases: long straight wires, long solenoids, and conductive slabs or cylindrical conductors carrying [current density](/ap-physics-c-e-m/key-terms/current-density "fv-autolink").
- Use [superposition](/ap-physics-c-e-m/key-terms/superposition "fv-autolink") to add the vector fields from multiple current sources.
- Ampere's law is Maxwell's fourth equation; with Maxwell's addition, a changing electric field also creates a magnetic field, but you are not expected to calculate with that term.

## Magnetic Fields from Moving Charges

### Ampere's Law and Magnetic Fields

Ampere's law ties electric currents to the magnetic fields they create. It is written as a line integral around a closed path:

$$\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{enc}$$

- $\vec{B}$ is the magnetic field along the closed path, and $I_{enc}$ is the current passing through the area bounded by that path.
- $\mu_0$ is the [permeability of free space](/ap-physics-c-e-m/key-terms/permeability-of-free-space "fv-autolink"), $4\pi \times 10^{-7}$ T·m/A.
- Only current that pierces the loop counts. Currents outside the loop do not add to $I_{enc}$, even though they still affect $\vec{B}$ at points along the loop.

For a long, straight current-carrying wire, Ampere's law gives:

$$B_{\text{wire}} = \frac{\mu_0}{2\pi} \frac{I}{r}$$

- $B_{\text{wire}}$ is the field strength a distance $r$ from the wire.
- The field forms concentric circles around the wire, with direction set by the right-hand rule (point your thumb along the current, and your fingers curl in the field direction).

For a long ideal solenoid, Ampere's law gives:

$$B_{\text{sol}} = \mu_0 n I$$

- $n$ is the number of turns per unit length.
- The field inside is uniform and parallel to the axis, and the field outside is treated as negligible. Unless told otherwise, assume solenoids are very long.

### Choosing an Amperian Loop

An Amperian loop is an imaginary closed path you draw to apply Ampere's law. Picking it well is what makes the math easy.

- The loop is a closed path around a current-carrying conductor.
- Choose the loop shape to match the symmetry of the current so that $\vec{B}$ is either constant and parallel to $d\vec{\ell}$ or [perpendicular](/ap-physics-c-e-m/unit-12/2-magnetism-and-moving-charges/study-guide/aujVCr641dSEbfts "fv-autolink") to it.
- When $\vec{B}$ is constant in magnitude and parallel to the path, the integral simplifies to $B$ times the path length.
- Segments where $\vec{B}$ is perpendicular to $d\vec{\ell}$ contribute zero to the integral.

A reliable approach:

1. Identify the current distribution and its symmetry.
2. Choose an Amperian loop that uses that symmetry.
3. Determine which parts of the loop actually contribute to the integral.
4. Find the enclosed current $I_{enc}$.
5. Solve for the unknown magnetic field.

For a cylindrical conductor or conductive slab carrying a current density, remember that $I_{enc}$ is only the current inside your loop. Inside the conductor, that may be a fraction of the total current.

### Superposition of Magnetic Fields

When several current-carrying conductors are present, the net magnetic field at a point is the vector sum of the individual fields.

- Each current source produces its own field.
- Add the fields as vectors: $\vec{B}_{total} = \vec{B}_1 + \vec{B}_2 + \vec{B}_3 + \dots$
- Direction matters as much as magnitude, so track field directions carefully with the right-hand rule.

Superposition applies to setups like multiple parallel wires, combinations of loops and straight segments, and current distributions in conductors.

### Maxwell's Equations and Electromagnetism

Ampere's law is one of Maxwell's four equations describing electromagnetism. Maxwell extended it to include the effect of a changing electric field:

- Ampere's law: $\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I$
- With Maxwell's addition: $\oint \vec{B} \cdot d\vec{\ell} = \mu_0 I + \mu_0 \varepsilon_0 \dfrac{d\Phi_E}{dt}$

The term $\mu_0 \varepsilon_0 \dfrac{d\Phi_E}{dt}$ shows that a changing electric field generates a magnetic field, much like a moving [charge](/ap-physics-c-e-m/unit-10/2-redistribution-of-charge-between-conductors/study-guide/3zelmsMupFfJh7VP "fv-autolink") does. This completes the link between electricity and magnetism. For the exam, understand this idea conceptually, but you are not expected to calculate using the changing-electric-field term.

> **Boundary Statement**
>
> The exam only expects quantitative application of Ampere's law for situations with symmetrical magnetic fields, such as long straight wires, long current-carrying solenoids, and conductive slabs or cylindrical conductors carrying a current density. You are not expected to use Maxwell's fourth equation with a changing electric field, but you should understand that a changing electric field generates a magnetic field.

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

### Problem Solving

- Start by checking the symmetry. If the field is not symmetric enough for a clean loop, Ampere's law will not isolate $B$, and you may need the Biot-Savart law instead.
- Write the line integral, then use symmetry to pull $B$ out: for a circular loop concentric with a wire, $\oint \vec{B} \cdot d\vec{\ell} = B(2\pi r)$.
- Set that equal to $\mu_0 I_{enc}$ and solve. Keep units consistent and carry $\mu_0 = 4\pi \times 10^{-7}$ T·m/A.

### Free Response

- Derive symbolic expressions before plugging in numbers when the question asks for a derivation.
- For a current density problem, write $I_{enc}$ as an integral of $J$ over the area inside the loop, then solve for $B$ both inside and outside the conductor.
- When you state a field direction, justify it with the right-hand rule, not just by naming the law.

### Common Trap

- Do not confuse $I_{enc}$ with the total current. Only current passing through the loop counts.
- Watch the solenoid formula: $n$ is turns per unit length, not the total number of turns.

## Common Misconceptions

- Ampere's law does not say $\vec{B}$ is zero where $I_{enc}$ is zero. It only says the line integral around the loop is zero. The field at points on the loop can still be nonzero due to outside currents.
- The field of a long straight wire falls off as $1/r$, not $1/r^2$. That $1/r^2$ behavior belongs to a single small current element in the Biot-Savart law, not the whole wire.
- Ampere's law is always true, but it is only useful for finding $B$ when there is enough symmetry. For asymmetric cases, you still need Biot-Savart.
- The field outside an ideal solenoid is treated as negligible, not because no field exists, but because the long-solenoid model approximates it as zero.
- In the solenoid formula, $B$ does not depend on the radius or on where you are inside, as long as you are well inside a long solenoid. The field is uniform.

## Practice Problem 1: Magnetic Field of a Wire

> A long, straight wire carries a current of 5.0 A. Calculate the magnetic field strength at a distance of 10 cm from the wire.

**Solution**

Apply the formula for the magnetic field around a long, straight wire derived from Ampere's law:

$$B_{\text{wire}} = \frac{\mu_0}{2\pi} \frac{I}{r}$$

Given:
- Current $I = 5.0$ A
- Distance $r = 10$ cm = 0.10 m
- Permeability of free space $\mu_0 = 4\pi \times 10^{-7}$ T·m/A

Substituting these values:

$$B_{\text{wire}} = \frac{4\pi \times 10^{-7}}{2\pi} \frac{5.0}{0.10}$$

$$B_{\text{wire}} = 2 \times 10^{-7} \times \frac{5.0}{0.10}$$

$$B_{\text{wire}} = 2 \times 10^{-7} \times 50$$

$$B_{\text{wire}} = 1.0 \times 10^{-5} \text{ T or } 10 \text{ μT}$$

The magnetic field at 10 cm from the wire is 10 μT, directed tangentially around the wire according to the right-hand rule.

## Practice Problem 2: Solenoid Magnetic Field

> A solenoid has 200 turns per meter and carries a current of 3.0 A. What is the magnetic field strength inside the solenoid?

**Solution**

For a long solenoid, use the formula derived from Ampere's law:

$$B_{\text{sol}} = \mu_0 n I$$

Given:
- Number of turns per unit length $n = 200$ turns/m
- Current $I = 3.0$ A
- Permeability of free space $\mu_0 = 4\pi \times 10^{-7}$ T·m/A

Substituting these values:

$$B_{\text{sol}} = 4\pi \times 10^{-7} \times 200 \times 3.0$$

$$B_{\text{sol}} = 4\pi \times 10^{-7} \times 600$$

$$B_{\text{sol}} = 7.54 \times 10^{-4} \text{ T or } 0.754 \text{ mT}$$

The magnetic field inside the solenoid is 0.754 mT, directed along the axis of the solenoid.

## Related AP Physics C: E&M Guides

- [12.2 Magnetism and Moving Charges](/ap-physics-c-e-m/unit-12/2-magnetism-and-moving-charges/study-guide/aujVCr641dSEbfts)
- [12.1 Magnetic Fields](/ap-physics-c-e-m/unit-12/1-magnetic-fields/study-guide/hK7HuQSBBDlQOdh5)
- [12.3 Magnetic Fields of Current-Carrying Wires and the Biot-Savart Law](/ap-physics-c-e-m/unit-12/3-magnetic-fields-of-current-carrying-wires-and-the-biot-savart-law/study-guide/9L5jxtAFTJoI6u5v)

## Vocabulary

- **Amperian loop**: A closed imaginary path drawn around a current-carrying conductor used to apply Ampère's law.
- **Ampère's law**: A fundamental equation that relates the magnitude of the magnetic field to the electric current enclosed by a closed imaginary path.
- **Maxwell's addition**: The modification to Ampère's law stating that a changing electric field creates a magnetic field in addition to the magnetic field created by electric current.
- **Maxwell's equations**: A collection of four fundamental equations that fully describe electromagnetism and the behavior of electric and magnetic fields.
- **changing electric field**: A time-varying electric field that generates a magnetic field according to Maxwell's equations.
- **conductive cylinder**: A cylindrical conductor carrying current to which Ampère's law can be applied to determine the magnetic field.
- **current density**: The amount of electric current flowing per unit cross-sectional area of a conductor; a vector quantity.
- **current-carrying wire**: A conductor through which electric current flows, creating a magnetic field in the surrounding space.
- **electric current**: The flow of electric charge through a conductor, which generates a magnetic field.
- **magnetic field**: A vector field that determines the magnetic force exerted on moving electric charges, electric currents, or magnetic materials.
- **moving charge carrier**: An electric charge in motion that generates a magnetic field around it.
- **principle of superposition**: The principle that the total electric field is the vector sum of fields produced by individual charges.
- **solenoid**: A coil of wire wound in a tight, helical pattern that produces a uniform magnetic field when current flows through it.

## FAQs

### What is Ampère's law in AP Physics C: E&M?

Ampère's law relates the line integral of the magnetic field around a closed path to the current enclosed by that path: ∮B · dℓ = μ0 I_enc.

### What is an Amperian loop?

An Amperian loop is an imaginary closed path chosen around a current distribution so Ampère's law can use symmetry to solve for the magnetic field.

### When is Ampère's law useful for finding magnetic fields?

Ampère's law is most useful when symmetry makes the magnetic field constant or perpendicular along parts of the loop, such as long wires, long solenoids, slabs, or cylinders.

### What magnetic field does Ampère's law give for a long straight wire?

For a long straight wire, Ampère's law gives B = μ0 I / (2πr), with direction set by the right-hand rule.

### What magnetic field does Ampère's law give inside a long solenoid?

For a long ideal solenoid, Ampère's law gives B = μ0 n I inside, with a nearly uniform field and negligible outside field.

### What is the AP Physics C boundary for Ampère's law?

The exam limits quantitative Ampère's law applications to symmetric magnetic fields and does not require calculations using Maxwell's changing-electric-field term.

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