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The force on a moving charged particle in a magnetic field is F = q(v × B), perpendicular to both velocity and field. This force does no work, changing only direction, not speed. The magnitude is F = qvB sin θ. This is the fundamental mechanism by which magnetic fields deflect moving charges.
From your study of the cross product, you know that v × B produces a vector perpendicular to both v and B, with magnitude |v||B|sinθ where θ is the angle between them. The Lorentz magnetic force F = q(v × B) is exactly this cross product scaled by the charge q. The direction follows the right-hand rule: point fingers along v, curl toward B, and the thumb points in the direction of the force on a positive charge. For a negative charge, the force is reversed.
The most striking feature of the magnetic force is that it does no work. Work requires a force component along the direction of motion, but F = q(v × B) is always perpendicular to v by definition of the cross product. If no work is done, the kinetic energy (and therefore speed) cannot change. The magnetic force can only redirect a particle — it acts as a pure steering force. This has a remarkable consequence: a charged particle moving perpendicular to a uniform magnetic field undergoes uniform circular motion. The magnetic force provides the centripetal acceleration, giving mv²/r = qvB, so the orbital radius is r = mv/(qB). Faster particles and heavier particles orbit in larger circles; stronger fields produce tighter orbits.
The magnitude formula F = qvB sinθ tells you that the force is maximum when v ⊥ B (sinθ = 1) and zero when v ∥ B (sinθ = 0). A particle moving exactly parallel to the field experiences no magnetic force at all — only when it has a component of velocity perpendicular to B does the force appear. This selectivity makes the Lorentz force geometrically sensitive in ways that an electric force is not.
These principles underlie technologies from mass spectrometers (where radius r = mv/qB separates ions by mass-to-charge ratio) to particle accelerators (where magnetic fields bend high-energy beams around circular tracks) to the aurora borealis (where Earth's magnetic field funnels charged solar wind particles toward the poles). The combination of the electric and magnetic contributions, F = q(E + v × B), is the complete Lorentz force law, unifying how charged particles respond to all electromagnetic fields.