The Magnetic Field Around a Wire

Current in a wire makes a magnetic field that circles around it. Walk any loop around the wire and the field adds up to the same total, proportional to the current. Walk a loop that misses the wire and it adds up to zero. That's Ampère's law, and it hides a catch in Green's theorem.

Click or drag anywhere to start a new trajectory there.

How it works

Green's and Stokes' theorems

In one sentence

Green's theorem $\oint_C P\,dx + Q\,dy = \iint_D (Q_x - P_y)\,dA$ holds when $P$ and $Q$ have continuous partial derivatives on the region $D$ enclosed by $C$ ; the field $\frac{(-y, x)}{x^2 + y^2}$ fails this at the origin and circulates $2\pi$ around it.

General formula

{\htmlData{sym=0}{\mathbf B}} = \frac{\htmlData{sym=1}{{I}}}{x^2 + y^2}\,\big(-y,\; x\big), \qquad {\htmlData{sym=2}{\oint_C}} {\htmlData{sym=0}{\mathbf B}}\cdot {\htmlData{sym=3}{d\mathbf r}} = \begin{cases}2\pi {\htmlData{sym=1}{I}} & \text{if } C \text{ circles the wire}\\ 0 & \text{if not}\end{cases}

Away from the wire the field is curl-free, so small loops there sum to nothing. Every loop around the wire can be shrunk to a tiny circle around it, and that circle always sums to $2\pi I$ .

What the symbols mean

$\mathbf B$
Magnetic field
Circles the wire, weaker farther away.
$I$
Current
Amps flowing in the wire, which points out of the screen at the origin.
$\oint_C$
Around the loop
Add up B along the whole closed path C.
$d\mathbf r$
Path step
A tiny step along the loop.

The field circles the wire and weakens like $1/r$ ; its curl is zero everywhere except at the wire itself.
Green's theorem says $\oint_C \mathbf B\cdot d\mathbf r = \iint_D \operatorname{curl}\mathbf B\,dA$ , so a loop that misses the wire gives 0.
A loop around the wire gives $2\pi I$ no matter its shape or size, even though the curl is zero at every point you can integrate over.
There's no contradiction: Green's theorem needs the field to be smooth everywhere inside, and this one blows up at the wire. In 3D this is Stokes' theorem and Ampère's law.

Learn the math

Conservative Vector FieldsCalc III · Lesson 43, Line Integrals Green's TheoremCalc III · Lesson 44, Line Integrals Stokes' TheoremCalc III · Lesson 49, Surface Integrals The Biot–Savart LawE&M · Lesson 18, Magnetic Fields and Forces Ampère's LawE&M · Lesson 19, Magnetic Fields and Forces The Displacement CurrentE&M · Lesson 26, Maxwell's Equations and Light Maxwell's EquationsE&M · Lesson 27, Maxwell's Equations and Light

Learn the math

Conservative Vector FieldsCalc III · Lesson 43, Line Integrals

Green's TheoremCalc III · Lesson 44, Line Integrals

Stokes' TheoremCalc III · Lesson 49, Surface Integrals

The Biot–Savart LawE&M · Lesson 18, Magnetic Fields and Forces

Ampère's LawE&M · Lesson 19, Magnetic Fields and Forces

The Displacement CurrentE&M · Lesson 26, Maxwell's Equations and Light

Maxwell's EquationsE&M · Lesson 27, Maxwell's Equations and Light