Force on a Current-Carrying Wire

A moving charge in a magnetic field feels a force F = qv x B. A wire carrying current is just a tube full of charges all drifting together, so the wire as a whole feels the sum of all those tiny forces. Add them up over a straight segment of length L carrying current I, and the messy sum of individual qv x B forces collapses into one clean vector equation: F = I L x B, where L points along the wire in the direction of conventional current. The magnitude is F = BIL sin(theta), with theta the angle between the wire and the field; the force is maximum when the wire is perpendicular to B and zero when it runs parallel to B.

The direction always comes from the cross product, so the right-hand rule tells you which way the wire jumps: point your fingers along the current, curl toward B, and your thumb gives F. This is exactly why a current-carrying wire jerks sideways near a magnet — the field is pushing on every charge inside it at once.

For a curved wire, you cannot just multiply BIL; you have to integrate I dL x B along the path, or use the shortcut that the net force on any wire only depends on the straight-line vector from start to end, not the actual path shape, as long as B is uniform. That shortcut is why a closed loop feels zero net force in a uniform field: start and end are the same point, so the closed integral of dL is the zero vector. The loop can still spin, because torque depends on where each force acts, not just the total force — a bridge to how loops behave as magnetic dipoles next.