Ampere's Law Equation, Derivation & Rearrangements

Core idea

Overview

Relates magnetic circulation around a closed path to enclosed steady current. It is content-only because the displayed integral or circulation law is a decision rule before choosing a specific geometry.

When to use: Use this to decide which source, path, or geometry controls the magnetic or induced-electric field.

Why it matters: It explains where the simpler magnetic-field formulas in this batch come from.

Walkthrough

Derivation

Derivation of Ampere Law

Relates magnetic circulation around a closed path to enclosed steady current.

The path and surface orientation are chosen consistently.
The electromagnetic fields are described by classical electromagnetism.

1

Read the law

Identify the field circulation, source, and sign convention.

\oint B \cdot d l = μ_{0} I_{e n c}

2

Match geometry

Only then can the law be reduced to a scalar formula.

choose path/surface/current element

Result

choose path/surface/current element

Source: Moebs, Ling, and Sanny, University Physics Volume 2, OpenStax, 2016, chapter 12, accessed 2026-04-09

Free formulas

Rearrangements

Solve for $law$

Use the law as a geometry decision rule

draw geometry and choose orientation

Choose the correct path, surface, or current element before reducing the relation.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

Contains advanced operator notation (integrals/sums/limits)

Why it behaves this way

Intuition

Ampere's law ties the circulation of B around a closed path to the enclosed steady current.

\oint

closed-loop integral

Integration taken all the way around a closed path.

B

magnetic field

The field being circulated around the loop.

d l

path element

A tiny segment along the chosen loop.

μ_{0}

vacuum permeability

The magnetic constant that scales the relation.

I_{e n c}

enclosed current

The current passing through the loop's interior.

Signs and relationships

=: The circulation equals permeability times enclosed current.

Free study cues

Insight

Canonical usage

Ampere's Law is used to calculate the magnetic field strength around a current-carrying conductor by integrating the magnetic field along a closed path and relating it to the enclosed current.

Common confusion

Students may confuse the enclosed current ( $I_{e}$ nc) with the total current flowing in the circuit, or incorrectly define the closed path (∮ dl).

Dimension note

This equation involves quantities with physical units and is not inherently dimensionless.

Unit systems

$B$ T · Magnetic field strength (Tesla).

dlm · Infinitesimal element of the closed path length (meter).

$I_{e} n c$ A · Enclosed electric current (Ampere).

One free problem

Practice Problem

In the integral expression for Ampere's Law, what does the circle on the integral sign signify regarding the path of integration?

Solve for: $\oint B \cdot d l$

Hint: Consider the geometric requirement for a circulation law.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In These laws are used to derive fields of wires, loops, solenoids, and induction setups, Ampere's Law is used to calculate \oint\vec B\cdot d\vec l from the measured values. The result matters because it helps check whether a circuit component is operating within the required voltage, current, power, or resistance range.

Study smarter

Tips

Draw the loop or current element first.
Use the sign/orientation convention consistently.
Choose a simpler derived formula only after matching the geometry.

Avoid these traps

Common Mistakes

Treating an integral law as a one-line scalar calculator.
Ignoring path orientation or enclosed current/flux.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Relates magnetic circulation around a closed path to enclosed steady current.

Use this to decide which source, path, or geometry controls the magnetic or induced-electric field.

It explains where the simpler magnetic-field formulas in this batch come from.

Treating an integral law as a one-line scalar calculator. Ignoring path orientation or enclosed current/flux.