Magnetic field and the Lorentz force

In 1820 Oersted saw a compass needle swing when a current flowed in a nearby wire. The needle sets itself along the tangent to a circle centred on the wire; reversing the current reverses it, and a bigger current or a closer needle deflects it more.

Iron filings round a current-carrying wire settle into concentric circles. Oersted's conclusion: moving charges, that is currents, set up a magnetic field in the space around them.

Drawing convention: a dot (⊙) is a current or field coming out of the page, a cross (⊗) one going into it, like the tip and the tail feathers of an arrow.

The magnetic field B is a vector field, defined at every point, and it obeys superposition: the fields of several sources add as vectors, just as electric fields do.

A charge q moving with velocity v through fields E and B feels the Lorentz force F = q[E + v × B]. The electric part does not care about motion; the magnetic part does.

The magnetic force q(v × B) is zero for a charge at rest and zero when v is parallel or antiparallel to B. It is perpendicular to both v and B, with direction from the right-hand (screw) rule, and it reverses for a negative charge.

Its size is qvB sin θ, where θ is the angle between v and B. This defines the unit: 1 tesla (T) is the field that pushes 1 N on 1 C moving at 1 m/s at right angles to it, so 1 T = 1 N s C⁻¹ m⁻¹.

The tesla is large. The non-SI gauss is 10⁻⁴ T, and the earth's field is about 3.6 × 10⁻⁵ T.