The Electrostatic Field and Electrostatic Potential Relationship

By Dr. Eugene Patronis

This is the fourth in a series of articles dealing with coaxial cables operating in the frequency span from direct current through the microwave region. The first article, which dealt only with currents, inductance, and magnetic fields in coaxial cables, was intended to be a standalone article directed toward a readership that was assumed to be familiar with the mathematics and physics of scalar and vector fields. Subsequently, the author was urged to cover both the electric as well as magnetic properties of the cable from both a circuit as well as field point of view. In order to accomplish this in a meaningful way, now for a wider readership, the second article was entitled “Mathematics Primer for Vector Fields”. This article treated the general mathematics of vectors as well as vector and scalar fields and concluded with the introduction of the gradient theorem of a scalar field. The third article in the series was entitled “Gauss’s Law and the Electrostatic Field” in which the divergence theorem of vector analysis plays a paramount role. This article concluded with the calculation of the electrostatic potential difference as well as the capacitance between the center and outer conductors of a section of charged coaxial cable. The first order of business in the present article is to justify this calculation. In doing so it is necessary to introduce another theorem of vector analysis known variously as the curl, circulation, or Stoke’s theorem.