Magnetic Lake: Dipole Field & Ferro-Surface Ripple (2D)

This 2D companion replaces the original's decorative particle swirl with an actual physics engine: the exact 3D magnetic-dipole formula sliced through a plane, B(r) = (3(m·r̂)r̂ − m) / r³, drives a live compass-needle grid — every needle re-aligns instantly when the magnet is dragged or rotated, exactly like real compasses over a bar magnet. The same field's squared magnitude drives a damped 2D wave equation coupled to a magnetophoretic-pressure target, so the lake's surface visibly ripples whenever the magnet moves, rotates, or its strength changes — every ripple traces back to a real field computation, not a canned animation. Four independent sliders (strength, orientation, wave speed, damping) and a draggable magnet let you test the physics directly: flip the polarity slider and every needle reverses, raise the wave speed and ripples race outward faster, raise damping and they die out sooner.