A bar magnet floats under a still lake. Its field is the exact magnetic-dipole formula sliced through the horizontal plane it lies in:
B(r) = (3(m·r̂)r̂ − m) / r³
Every needle in the grid points along B at its own location — drag the magnet or rotate it and every needle re-aligns instantly, exactly like real compasses over a bar magnet. The lake's ferromagnetic floor feels a magnetophoretic pressure proportional to B² (the same ∇B² law that pulls real ferrofluid toward strong field), which the surface chases through a damped 2D wave equation:
hẁ = k·(B²·p − h) − γ·ḣ + c²·∇²h
Move the magnet fast and the target pressure field changes faster than the surface can track — that mismatch is what launches the visible ripples, exactly the way a real disturbance to a field source radiates a wave. Wave speed (c) and damping (γ) are independent physical knobs on that same PDE.
- Strength — dipole moment magnitude; scales both needle deflection and the depth of the magnetic dimple.
- Orientation — rotates the dipole vector; watch the two field lobes swing with it.
- Wave speed / damping — surface tension and viscosity of the wave equation driving the ripples.