Maxwell equations for a charging capacitor

The charging capacitor is used as a standard paradigm for illustrating the concept of the Maxwell "displacement current". A certain aspect of the problem, however, is often overlooked. It concerns the conditions for satisfaction of the Faraday-Henry law both in the interior and the exterior of the capacitor. In this article the situation is analyzed and a mathematical process is described for obtaining expressions for the electromagnetic field that satisfy the full set of Maxwell’s equations both inside and outside the capacitor. (Note: This is an extended version of  the article "Some remarks on the charging capacitor problem", to be published in Advanced Electromagnetics. It is written mostly for educational purposes but contains new results.)