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Biot-Savart 3D Field Explorer

Run a current around a loop of wire and it wraps the space around it in a magnetic field. The Biot-Savart law adds up the contribution of every little length of wire, and for a circular loop the result is the familiar pattern: field lines thread up through the middle, flare out, and curl back around the outside, exactly like a bar magnet. The shape is fixed by the geometry; the current only sets the strength. Stack the loops cleverly and the field becomes useful. Two loops a radius apart, a Helmholtz pair, make a remarkably uniform field in the gap between them; many loops in a tube, a solenoid, make a nearly perfectly uniform field down the whole inside and almost nothing outside, which is how every electromagnet works. The scene shows a slice through the axis with the field lines and their strength; the diagnostic plots the field along the axis, where the loop's peak, the Helmholtz flat spot, and the solenoid's plateau each show up.

Figure 1. Biot-Savart magnetic field of current coils. Top: a slice through the axis showing field lines (with strength as a colour map) and the wire crossings (current into and out of the plane). Bottom: the axial field Bz along the symmetry axis, with the single-loop peak, the Helmholtz flat region, or the solenoid plateau. Method: Biot-Savart summation over discretized current segments.
coilloop
current I1.5
radius R1.00
separation1.00

WHAT TO TRY

  • Start with a single loop and see the bar-magnet pattern: lines up through the middle, around the outside, back in.
  • Switch to the Helmholtz pair and slide the separation: at d = R (separation equals radius) the axial field flattens into the uniform Helmholtz plateau; pull the coils apart and a dip opens between them (the curve splits into two humps).
  • Switch to the solenoid and stretch its length: the lines lock parallel down the inside and the plateau in the lower plot grows longer and flatter, near zero outside.
  • Change the radius R: a wider coil spreads the field and lowers the central strength; a tighter coil concentrates it.
  • Raise the current I: the field pattern keeps its shape (the geometry sets that) but the strength scales up and the field-line arrows speed up and brighten.