Fundamentally Maxwell’s equations describe the origins of electric and magnetic fields. Given a set of conditions on the right hand side of the equations, you’ll have fields described by the left hand side. Between the four equations the fields are uniquely specified, and there is nothing more to include. Thus far we’ve seen electric fields generated by electric charge, electric fields generated by changing magnetic fields, and magnetic fields not generated by magnetic charge because there isn’t any. Conspicuously absent is any way to generate magnetic fields. It is this fourth equation that closes the circle and describes the origin of the magnetic field.
The left hand side is the curl of the magnetic field, precisely analogous to yesterday’s curl of the electric field. The right hand side contains two terms, for there are two ways to generate magnetic field circulation. The first is via an electric current, with current density given by J. The electromagnets that heft huge piles of metal scrap at junkyards work on this principle. Run current through a wire, and the wire will develop a magnetic field surrounding it. Via Wikipedia:
The second term on the right is of the same type as in the equation for the curl of the electric field. In this equation, a changing electric field produces a magnetic field of specified curl. This is entirely without reference to any charge. While the J term creates a magnetic field in the presence of flowing charge, it’s not required. So long as there’s a changing electric field, the derivative on the right will be nonzero and you have yourself a magnetic field.
Historically the equation with only the first term on the right is called Ampere’s Law. This limited equation turned out to fail in explaining a large number of physical phenomena, such as the magnetic field between plates of a charging capacitor. Between the plates there is no current, but there is a field. It was Maxwell who first correctly accounted for this, wrote the complete equation, and worked out the consequences of the four combined equations that now bear his name.
The four equations are almost symmetric in their form, with the symmetry broken only by the absense of magnetic charge. If it existed a magnetic charge density in the magnetic divergence equation and a magnetic current density in the electric curl equation would complete the symmetry. This sat as a somewhat aesthetically unsatisfactory condition for many years, despite the fantastic explanatory power of the theory of electromagnetism. It took another brilliant physicist to (mostly) put this enduring question to bed. Though the classical mechanics of Newton required relativistic correction, Einstein showed that Maxwell’s theory was already consistent with the theory of special relativity. Electricity and magnetism turn out to be literally the same phenomenon in different reference frames. Rewritten in a relativistically natural way (the term of art is covariant), the broken symmetry vanishes and the slight lopsidedness of the equations as we’ve seen them vanish into the ether. Well, as it were.
All the countless thousands of pages of classical electromagnetism textbooks are lagniappe following from these four equations. The consequences of those equations are numerous and profound, and of course those textbooks are indispensable as no one is clever enough to find all of their subtleties for themselves. In fact, we have four equations and five days in the working week. It would be a shame to waste the last. Tomorrow we’ll look at the most profound of the implications of these equations; the one Maxwell is most famous for discovering, the one we see every time we see.