# Electric & Magnetic Field Line Plotter

> Field lines and equipotentials of point charges and current-carrying wires, with probe values. Place point charges (in nC) and long straight wires (in A) in a plane and get a drawing of the electric field lines from positive to negative charges, the magnetic field loops around the wires and the equipotential lines, traced numerically from Coulomb’s law and the Biot–Savart law. The number of lines per charge follows Gauss’s law; probe points give the exact field strength, direction, potential and magnetic flux density. Part of the Proofline math notebook.

Part of usefulweb.app (https://usefulweb.app/). Part of [Proofline](https://proofline.usefulweb.app/) (ownerApp: proofline). Results are deterministic calculations from the inputs and the listed assumptions; the API returns exactly what the web page shows.

## Use it
- Human interface: https://prooflinefieldlines.usefulweb.app/ (interactive tool at https://prooflinefieldlines.usefulweb.app/tool)
- JSON API: POST https://prooflinefieldlines.usefulweb.app/api/v1/calculate with a JSON object of inputs (schema: https://prooflinefieldlines.usefulweb.app/api/v1/tool)
- OpenAPI: https://prooflinefieldlines.usefulweb.app/openapi.json
- MCP (Streamable HTTP): https://prooflinefieldlines.usefulweb.app/mcp - tool `proofline_field_lines`

## Inputs
- sources (textarea, required): Charges and wires (one per line) - “charge q x y” is a point charge q in nC at (x, y) in cm; “wire I x y” is a long straight wire carrying I amperes out of the page (negative: into the page). At most 16 lines; # starts a comment.
- linesPerUnit (number): Field lines per nC (or per 2 A)
- equipotentials (boolean): Draw equipotential lines
- extent (number): View half-height (cm) - Empty: fit the view around the sources.
- probes (textarea): Probe points (optional) - One point per line: x and y in cm. The table lists E, the potential and B there (at most 20).

## Method
- Electric field of point charges: E = Σ k·qᵢ·(r − rᵢ)/|r − rᵢ|³ with k = 1/(4πε₀) = 8.9875517923 × 10⁹ N·m²/C²; potential V = Σ k·qᵢ/|r − rᵢ|. Magnetic field of a long straight wire: |B| = μ₀·I/(2π·r), circling the wire anticlockwise for current out of the page (μ₀ = 1.25663706212 × 10⁻⁶ N/A²). Fields superpose.
- Field lines start on a small circle around each positive charge, evenly spaced in angle, in a number proportional to the charge, and follow the unit field direction with fixed RK4 steps until they reach a negative charge or leave the view. Negative charges also send lines backwards; those that reach a positive charge are dropped as duplicates. Magnetic lines start at radii of equal flux steps and close on themselves.
- Equipotentials are contour lines (marching squares) of V on a 161 × 121 grid at the potentials of a lone charge at fixed fractions of the view size, both signs, and V = 0.
- Verified: E = kq/r² of a single charge (1 nC at 1 cm: 89 875.5 V/m), E = 2kq/(d/2)² midway in a dipole, B = μ₀I/(2πr) (1 A at 1 cm: 20 µT), radial lines of a single charge, and line counts in proportion to the charges (Gauss’s law).

## Limits
- An idealised 2-D picture: point charges in vacuum and infinitely long wires. The line density shows the field strength only qualitatively in a plane drawing of a 3-D field.