Section 5.1 · Topic 3 of 6
5.1.3 Counting the unknowns
A quick count seems to show that the system is not closed. The four vector fields , , and have components, while the two curl equations give scalar equations and the two divergence equations : twelve unknowns and eight equations. The missing information is supplied by the constitutive relations of the medium, introduced in Section 3.4 and in Chapter 4,
which are six more scalar equations. That makes fourteen, apparently two too many. The resolution is that the two divergence equations are not independent of the two curl equations: as the box below shows, they follow from them together with the conservation of charge. The count is then twelve unknowns and twelve independent equations, and the system is closed. In a conductor a third relation, Ohm's law , ties the current density to the field.
Take the divergence of the first equation of Eq. (5.1.1). Since the divergence of a curl vanishes identically (Section 2.2),
so cannot change in time. If it was zero at any one instant (for instance before the sources were switched on) it is zero for ever, which is ④. Take the divergence of the second equation in the same way:
Charge conservation, the continuity equation of Section 5.4, states , so : if Gauss's law held once, it holds always, which is ③. The divergence equations are therefore initial conditions that the curl equations preserve.