The magnetic term has magnitude |q|vBsinθ. Use the right-hand rule for vec(v)×vec(B), then reverse the direction for negative charge. Magnetic field is measured in teslas, with 1 T=1 N/(A m).

Circular motion is a consequence, not a separate magnetic law

For velocity perpendicular to a uniform field and no other forces, the magnetic force supplies the centripetal acceleration:

The force is always perpendicular to instantaneous velocity. Consequently P=vec(F)·vec(v)=0: magnetic force alone changes direction, not kinetic energy. A velocity component parallel to the field remains unchanged, producing a helix rather than a circle.

For a proton moving at 1.0×10⁶ m/s perpendicular to 0.20 T, using mp=1.67×10⁻²⁷ kg and e=1.60×10⁻¹⁹ C, the radius is 0.0522 m, about 5.2 cm. Doubling speed doubles radius, whereas doubling field halves it. The period 2π r/v=2π m/(|q|B) is independent of speed in this nonrelativistic idealization.

A wire contains many moving charges

Summing magnetic forces over charge carriers in a straight wire segment gives

The vector vec(L) points along conventional current. A 0.10 m segment carrying 5.0 A perpendicular to 0.20 T feels 0.10 N. Opposite sides of a current loop can experience equal and opposite forces, giving zero net force but a torque. This is the basic motor geometry; a loop's magnetic moment is vec(μ)=IAn̂, and its torque is vec(τ)=vec(μ)×vec(B).

How can a motor do mechanical work if magnetic force does no work on a point charge? The motor is not an isolated charge in a static magnetic field. Its source drives current through electric fields, and forces on carriers are transmitted to the moving conductor. Track the entire electromechanical system: electrical input supplies mechanical output and losses. The point-charge perpendicular-force result has not failed.

Currents produce magnetic fields

For steady currents in vacuum, Ampère's law is

Around an effectively infinite straight wire, symmetry makes the field tangent to circles and equal in magnitude at fixed radius. Thus

With I=10 A and r=1.0 cm, this gives 2.0×10⁻⁴ T, or 0.20 mT. Curl the right-hand fingers around the wire with the thumb along current to determine direction. Nearby return wires alter the resulting field by superposition.

When symmetry does not make Ampère's law easy to evaluate, the steady-current Biot–Savart law adds the contributions from individual wire elements:

Here r̂ points from the source element to the observation point. At the center of a circular loop of radius a, every element is perpendicular to r̂, every distance is a, and every contribution points along the loop axis. Integrating dℓ around circumference 2π a gives B=μ₀I/(2a). This is a useful contrast: the wire's shape enters a common law through geometry, rather than each shape requiring a new physical principle.

A long, tightly wound solenoid approximately adds many loop fields into a nearly uniform interior field B≈μ₀nI, where n is turns per unit length; end effects are neglected. Magnetic materials can strongly modify this relation, often nonlinearly. Permanent magnets also involve microscopic magnetic moments, not literally a hidden macroscopic coil.

Magnetic field lines have no starts or ends at magnetic charges; they may form closed loops or extend without bound. No isolated magnetic monopoles have been established. In changing-field problems Ampère's law needs Maxwell's additional electric-flux term. The steady-current expression is a deliberately limited model, just as electrostatic potential is limited when induction becomes important.