Optics, electromagnetic theory Definition

Force-free magnetic field

Builds on Magnetic field Lorentz force Ampère's circuital law Curl Cross product

Definition

Let $U \subseteq \R^3$ be open, and let $B \colon U \to \R^3$ be a $C^1$ magnetic field with $\nabla \cdot B = 0$ . Then $B$ is force-free if $(\nabla \times B) \times B = 0 \quad \text{on } U,$ where $\nabla \times$ is the curl and $\times$ the cross product. Equivalently, on the set where $B \neq 0$ there is a function $\alpha$ with $\nabla \times B = \alpha B$ . The field is linear force-free if $\alpha$ is a constant on $U$ .

Remarks

By Ampère's circuital law in magnetostatics, $\nabla \times B = \mu_0 J$ , so the condition says that the Lorentz force density $J \times B$ vanishes. Taking the divergence of $\nabla \times B = \alpha B$ and using $\nabla \cdot B = 0$ gives $B \cdot \nabla \alpha = 0$ where $\alpha$ is $C^1$ , so $\alpha$ is constant along each field line. Curl-free fields, with $\alpha = 0$ , are force-free; they are called potential fields.

Example: $B = (\sin z, \cos z, 0)$ on $\R^3$ has $\nabla \cdot B = 0$ and $\nabla \times B = (\sin z, \cos z, 0) = B$ . So it is linear force-free with $\alpha = 1$ ; its field lines are horizontal and turn steadily with height.

Non-example: $B = (-y, x, 0)$ has $\nabla \times B = (0, 0, 2)$ , and $(\nabla \times B) \times B = (-2x, -2y, 0)$ , which is nonzero away from the $z$ -axis.

Reading path

To understand this page from the ground up, read these 77 pages in order. The foundations come first.

Cross product concept
Electric charge concept
Electric field concept
Euclidean space object
Function concept
Function composition concept
Infimum concept
Magnetic field concept
Maxwell's equations concept
Parametrised surface concept
Supremum concept
Lorentz force statement
Fréchet derivative definition
Total derivative concept
Function of several variables concept
Vector-valued function concept
Jordan measure definition
Surface area concept
Tangent plane and normal vector of a surface concept
Differentiable function of several variables concept
Iterated integral definition
Limit of a function of several variables concept
Partial derivative concept
Parametrised curve concept
Vector field concept
Orientable surface concept
Surface integral concept
Gradient concept
Continuity in R^n definition
Differentiable function of several variables definition
Partial derivative definition
Curl concept
Line integral of a vector field concept
Flux through a surface concept
Differentiable on an open set definition
Double integral definition
Jacobian matrix definition
Chain rule in several variables statement
Gradient definition
Higher partial derivatives definition
Biot–Savart law statement
Fubini's theorem statement
Iterated derivative definition
Triple integral definition
Chain rule for a composition with a curve statement
C^k function characterised by continuous derivatives statement
Differential form definition
Parametrised curve definition
Vector field definition
Wedge product of forms definition
Closed curve definition
Rectifiable curve definition
Parametrised surface definition
Conservative vector field definition
Divergence definition
Curl definition
Tangent vector of a curve definition
Line integral of a scalar field definition
Line integral of a vector field definition
Tangent plane and normal vector of a surface definition
Regular curve definition
Green's theorem statement
One-form definition
Surface area definition
Arc length parametrisation definition
Divergence theorem on a rectangle in the plane statement
Surface integral definition
Exterior derivative definition
Line integral as an integral over the parameter interval statement
Orientable surface definition
Gradient theorem statement
Flux through a surface definition
Divergence theorem statement
Generalised Stokes theorem statement
Stokes' theorem statement
Ampère's circuital law statement
Force-free magnetic field definition

Tag	`5SYY633TJE`
URI	`m2://5SYY633TJE`
Kind	definition
Writing	W0 stub
MSC	78A30, 76W05
Defines	Force-free magnetic field
Uses	Magnetic field, Lorentz force, Ampère's circuital law, Curl, Cross product
Changed	3 November 2026

Force-free magnetic field

Definition

Remarks

Reading path

Further reading

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