---
title: "Stellarator coils and field lines Interactive Lab | Sandbox Physics"
description: "How do three-dimensional coils twist a field, and what do repeated crossings of one plane reveal?"
canonical_url: "https://sandboxphysics.com/labs/w7x-stellarator/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/w7x-stellarator/"
last_modified: "2026-11-14"
---

# Stellarator coils and field lines

**How do three-dimensional coils twist a field, and what do repeated crossings of one plane reveal?** Three-dimensional coils supply both toroidal and poloidal field components, so a field line winds around the tube as it follows the large ring. Repeated crossings of one plane reveal its finite computed pattern. Estimating rotational transform also requires a continuous angle about the moving magnetic axis and checks of step size and trace length.

Compare two Wendelstein 7-X current references, follow computed field lines, and inspect fixed-plane crossings and finite winding about the moving axis.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Choose currents and a starting point

Numerical current programme $0$ Start outward from axis $0.05\,\mathrm{m}$ Start above axis $0\,\mathrm{m}$ Requested full toroidal turns $64$

Non-planar $1.62\,\mathrm{MA}$ Planar $0\,\mathrm{MA}$

Effective currents include winding turns. New currents need a newly computed axis and trace.

Numerical window

Numerical window half-height $1.4\,\mathrm{m}$

This window is an integration limit, not a material wall. The bounded current programme joins two source references; it is a numerical input family.

Model assumptions

A traceable W7-X CAD single-filament model supplies the prescribed-current vacuum field; it is not an as-built coil survey. Precomputed references retain actual crossings and numerical checks. New currents require a new axis and trace. Finite winding does not prove a unique surface or predict particle transport, wall heat or fusion power. Displayed wire thickness does not enter the field calculation, and integration limits are not material walls.

[Read how three-dimensional coils create a twisting field](https://sandboxphysics.com/stories/instruments/w7x-stellarator/)

## Follow a magnetic line from its source coils

**Background** Charged particles gyrate as they move along a magnetic field. A stellarator uses external three-dimensional coils to make the field wind around both the large ring and its cross-section.

**Why it matters** A long three-dimensional line soon obscures itself. Recording repeated crossings of a fixed plane turns that path into a legible two-dimensional sequence.

- **Focus question**: How do current ratios, starting position and tracing length change an actual section record?
- **One-sentence intuition**: The line and crossings use the same coil field. Continuous signed angles about the moving computed axis preserve the winding direction.

### Core mathematical model

#### Add the source-coil fields

$$
\mathbf B(\mathbf x)=\frac{\mu_0}{4\pi}\sum_j I_j\oint\frac{d\boldsymbol\ell_j\times(\mathbf x-\mathbf r_j)}{|\mathbf x-\mathbf r_j|^3}
$$

The source supplies each filament curve and effective current, already including winding turns. Illustrative conductor thickness never enters the field calculation.

#### Integrate in that same field

$$
\frac{dR}{d\phi}=R\frac{B_R}{B_\phi},\qquad\frac{dZ}{d\phi}=R\frac{B_Z}{B_\phi}
$$

R is distance from the central vertical axis, Z is height and φ advances around the ring. Events are located on actual global half-planes. A small toroidal component or numerical limit stops the calculation while retaining completed events.

#### Retain signed continuous winding

$$
\iota_N=\frac{\Delta\theta_{\mathrm{unwrapped}}}{\Delta\phi},\qquad N=\frac{\Delta\phi}{2\pi}
$$

θ is measured about the moving computed axis, from outward radial towards upward. Positive φ is counterclockwise viewed from above. N is signed full toroidal turns. A finite winding estimate alone does not establish a unique invariant magnetic surface.

### Common difficulties

#### Is the moving dot a particle?

**Typical misconception** The marker follows an electron orbit and therefore measures particle loss.

**Better mental model** It replays numerical drawing points in order. Real particles also gyrate, drift and collide; the field-line calculation provides no loss fraction.

#### Must the field line close?

**Typical misconception** A closed section curve requires a line to return to its starting point after a few turns.

**Better mental model** The curve can be the slice of a magnetic surface. One line may wind over that surface without closing exactly on itself.

### Run the experiment

1. 01

   #### Compare traceable current references

   Choose Standard reference and Author-code reference. Inspect the effective non-planar and planar currents.

   **What to observe:** Both use the same 70 source coils. The default loads a numerical reference with recorded checks; computing again creates a separate single-run record.

2. 02

   #### Let new currents produce a new line

   Drag the gold current handle on a coil, then compute the trace.

   **What to observe:** The quick arrows are three actual field probes at the new currents. A new axis and crossings require integration; old paths are never reshaped by the slider.

3. 03

   #### Compare the two outer starting points

   Compare Outer starting point and Author outer start in the section view.

   **What to observe:** Every point is a retained crossing event. When the toroidal field becomes too small, the toroidal-angle method stops and preserves earlier crossings. This is a numerical stop, not particle loss.

4. 04

   #### Compare section conventions

   Switch between one global plane and the five repeating sections. Inspect event counts and angle labels.

   **What to observe:** The default reference completes 64 full turns: 320 crossings of five physical half-planes spaced a fifth of a turn apart are overlaid by rotational symmetry, with 64 returns to the same base plane. If the seed is off the chosen plane, the first hit is an entry. Winding uses actual signed toroidal angle, with no further factor of five.

5. 05

   #### Inspect short and terminated records

   Choose One full turn and Restricted numerical window.

   **What to observe:** Completed crossings and stopping reasons remain available. The window limits integration; leaving it is not a particle hitting a wall.

Physical conclusion

Finite trajectories provide inspectable geometry. Numerical checks support their winding estimates; particle transport requires additional physics.

## Related experiments and articles

How physics instruments measure · Reading path

### How physics instruments measure

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why stellarator coils bend in three dimensions**](https://sandboxphysics.com/stories/instruments/w7x-stellarator/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)