Force on a Moving Charge & on a Current-Carrying Conductor

Welcome to Module 5. In the previous module you spent your time computing the magnetic field B produced by various current distributions using the Biot-Savart law and Ampere's law. You also met the magnetic vector potential A. Throughout all of that work the magnetic field was the output — the thing you solved for. Now we turn the relationship around: given an externally produced B field, what force does it exert on the charges and currents that happen to be moving through it?

This is the bridge between magnetostatics and practically everything electromagnetic that moves — from the beam in a mass spectrometer to the armature in an electric motor. The key player is the Lorentz magnetic force, and it comes in two closely related forms: one for a single moving charge and one for a macroscopic current-carrying wire.

The magnitude of the magnetic force is F = qvB sinθ, where θ is the angle between the velocity vector v and the magnetic field vector B. Several things are worth noting immediately. First, if the charge is at rest (v=0) there is no magnetic force at all — magnetic fields only act on moving charges. Second, the force is maximum when v and B are perpendicular (θ=90°) and zero when they are parallel (θ=0°). Third, and most importantly, the direction of the force is perpendicular to both v and B, determined by the right-hand rule: point your fingers in the direction of v, curl them toward B, and your thumb points in the direction of F for a positive charge. For a negative charge the force points in the opposite direction.

A current-carrying wire is nothing more than an enormous number of charge carriers drifting through a conductor. If each carrier feels a force qv_d×B, and there are n carriers per unit volume moving with drift velocity v_d through a wire of cross-section A and length L, the total force on the wire segment is obtained by summing over all carriers. The product nqv_dA is exactly the current I, and the length L gives the direction of current flow. The result is the macroscopic force law:

Here L is a vector whose magnitude is the wire length and whose direction is the direction of conventional current flow. The magnitude is F = BIL sinθ, where θ is now the angle between the wire (current direction) and B. The right-hand rule works the same way. When the wire is perpendicular to the field the force is maximum (F=BIL), and when the wire is parallel to the field the force vanishes. This simple equation is the basis for loudspeakers, railguns, and every galvanometer movement.

It is tempting to memorize just the magnitudes, but the cross-product structure is what makes magnetic forces physically interesting. Because the force is always sideways to the motion, a magnetic field can steer a particle — bending its path into a circle — without ever adding or removing energy. This is why mass spectrometers use magnetic fields to separate ions by charge-to-mass ratio: every ion travels at the same speed (the field does not change it), but ions with different q/m values have different radii of curvature. Similarly, in a wire the sideways force on each carrier is transmitted to the lattice through collisions, producing a net mechanical force on the conductor as a whole.