Maxwell's equations are expressed in the language of vector calculus, so a significant portion of the previous chapters has been devoted to explaining vector calculus, not Maxwell's equations. For better or worse, that's par for the course, and it's going to happen again in this chapter. The old adage "the truth will set you free" might be better stated, for our purposes, as "the math will set you free." And that's the truth.
In mid-1820, Danish physicist Hans Christian Ørsted discovered that a current in a wire can affect the magnetic needle of a compass. His experiments were quickly confirmed by François Arago and, more exhaustively, by André Marie Ampère. Ampère's work demonstrated that the effects generated by the current, which defined a so-called "magnetic field" (labeled B in Fig. 4.1), were centered on the wire, were perpendicular to the wire, and were circularly symmetric about the wire. By convention, the vector component of the field had a direction given by the right-hand rule: if the thumb of the right hand were pointing in the direction of the current, the curve of the fingers on the right hand gives the direction of the vector field.
Other careful experiments by Jean-Baptiste Biot and Félix Savart established that the strength of the magnetic field was directly related to the current I in the wire and inversely related to the radial distance from the wire r.