Magnetic Field Line Sketcher

Physics Electricity and Magnetism • Magnetic Fields and Sources

Written by STEM Calculators Team Published January 17, 2026 Updated February 24, 2026

7. Magnetic Field Line Sketcher

Generates qualitative magnetic field lines from simple current/magnet sources using superposition. Field lines are tangent to the local \(\mathbf{B}\) and form closed loops.

Units: meters (m), amperes (A), tesla (T). This tool is qualitative (for intuition/visualization). Inputs accept 1e-3, pi, sqrt(2), sin(), cos(), tan(), ln(), log(), abs(). Use * for multiplication.

Sources + Controls

Tool: Source type:

Wire source

Current \(I\) (A):

Field around a straight wire is circular: \(\mathbf{B}\propto \dfrac{I}{r}\,\hat{\boldsymbol{\phi}}\).

Direction:

Right-hand rule flips the circulation direction.

Dipole (bar magnet) source

Moment \(|\mathbf{m}|\) (arb.):

Qualitative dipole field in the plane. Lines go from N to S outside.

Angle \(\theta\) (deg):

\(\theta=0^\circ\) points along +x. \(\mathbf{m}\) points from S to N.

Pole spacing \(a\) (m):

Used only for drawing N/S markers + seeding near poles (visual/qualitative).

Field-line sketch settings

Line density:

Higher density = more seeds + more lines near strong fields.

Step \(ds\) (m):

Smaller \(ds\) gives smoother curves (slower).

Max steps/line:

Stops long integrations / prevents infinite loops.

Direction arrows:

Shows local direction along field lines.

Show guides:

Axis/grid and source glyphs.

Ready

Canvas interaction:
• Pan = drag background
• Zoom = wheel/trackpad or pinch
• Add = click (Tool: Add source)
• Move = drag a source (Tool: Move)
• Delete = click a source (Tool: Delete)
• Probe = click to show \(\mathbf{B}\) at a point (Tool: Probe)

Steps

Add one or more sources, then click Sketch.

Field-line sketch

2D cross-section (\(x\)-\(y\)): sources, \(\mathbf{B}\) field lines, and direction

Legend

Wire: ⊙ (+z) or ⊗ (−z) at its position; field lines are circles around it
Dipole/bar magnet: N/S markers; lines bulge from N to S
Field lines are tangent to \(\mathbf{B}\) everywhere (qualitative)
Probe point shows \(\mathbf{B}\) vector direction and magnitude (relative)

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Theory

What magnetic field lines mean

A magnetic field line is a curve whose tangent direction matches the magnetic field:

\[ \frac{d\mathbf{r}}{ds}=\pm \frac{\mathbf{B}}{|\mathbf{B}|}. \]

The “\(\pm\)” means you can trace in either direction. The curve is geometric; it does not “start” or “end.”

Magnetic fields obey \(\nabla\cdot\mathbf{B}=0\), so field lines form closed loops (no magnetic monopoles).

Superposition

If multiple sources are present, the total field is the vector sum:

\[ \mathbf{B}(\mathbf{r})=\sum_i \mathbf{B}_i(\mathbf{r}). \]

The sketcher uses this principle at every point to decide the field-line direction.

Infinite straight wire (along \(\pm\hat{\mathbf{z}}\))

Around a long straight wire, the field is circular (azimuthal direction \(\hat{\boldsymbol{\phi}}\)):

\[ |\mathbf{B}|=\frac{\mu_0 I}{2\pi r},\qquad \mathbf{B}\parallel \hat{\boldsymbol{\phi}}. \]

In the \(x\)-\(y\) plane with the wire at the origin,

\[ \mathbf{B}(x,y)=\frac{\mu_0 I}{2\pi}\,\frac{(-y,\ x)}{x^2+y^2}. \]

Direction comes from the right-hand rule: thumb along current, fingers curl with \(\mathbf{B}\). Switching \((+\hat{z})\leftrightarrow(-\hat{z})\) reverses the circulation.

Dipole / bar magnet (qualitative)

A bar magnet far from the magnet behaves like a magnetic dipole with moment \(\mathbf{m}\) (points from S to N). A common far-field model is:

\[ \mathbf{B}(\mathbf{r})=\frac{\mu_0}{4\pi}\left(\frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r})}{|\mathbf{r}|^5}-\frac{\mathbf{m}}{|\mathbf{r}|^3}\right). \]

This calculator uses a simplified in-plane version to generate a qualitative “N to S” bulging pattern.

How the sketch is generated (numerical idea)

Field lines are computed by placing seed points near sources and integrating the streamline equation. The stepper uses a midpoint (RK2) update for stability:

\[ \mathbf{r}_{k+1}=\mathbf{r}_k + ds\,\hat{\mathbf{b}}(\mathbf{r}_k), \quad \hat{\mathbf{b}}=\frac{\mathbf{B}}{|\mathbf{B}|}. \]

Smaller \(ds\) gives smoother lines but takes more steps. “Line density” increases the number of seed points.

Notes

This is a qualitative sketcher: it emphasizes correct geometry (closed loops, tangency, superposition), not high-precision field magnitude.
Very close to a source, ideal formulas diverge; the integrator stops near singular regions.
For solenoids/university cases: the same streamline idea works once you provide a field model \(\mathbf{B}(\mathbf{r})\) for that geometry.

Steps

Field-line sketch

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