Video narration
Why does a compass beside a current-carrying wire turn? Its needle responds to a magnetic field around the wire-invisible, but real. The Biot-Savart law shows how to calculate that field from tiny current segments. For one segment: dB = (μ0/4π) I dl × r / r2. I is current; dl points along the segment; r points from it toward the compass, and r is the distance. The contribution grows with current, falls with distance squared, and depends on the sine of the angle between dl and r. Use the right-hand rule: point along dl, curl toward r; your thumb gives the field's direction. The wire's field isn't one unexplained effect: every tiny segment contributes, and those contributions add. Add them along the whole wire-the integral-and you have its total field at the compass. So what turns the needle? The combined field from all those tiny pieces.