CHAPTER 11 - THE ELECTROMAGNETIC FIELD EQUATIONS

Abstract

This chapter focuses on the electromagnetic field equations. From the first pair of Maxwell's equations, it was found that the two equations still do not completely determine the properties of the fields. This is clear from the fact that they determine the change of the magnetic field with time (the derivative ∂H/∂t) but do not determine the derivative ∂E/∂t. According to Gauss's theorem of electromagnetic fields, the net outward normal electric flux through any closed surface is proportional to the total electric charge enclosed within that closed surface. On the other hand, Stokes' theorem states that the line integral of an electric field (E) bounding over a closed surface is equal to curl integral of E over the surface. The integral of a vector over a closed contour is called the circulation of the vector around the contour. The circulation of the electric field is also called the electromotive force in the given contour. Thus, the electromotive force in any contour is equal to minus the time derivative of the magnetic flux through a surface bounded by this contour. The Maxwell equations can be expressed in four-dimensional notation. In addition to the Gaussian system, one also uses the Heaviside system, in which a = −¼. In this system of units the field equations have a more convenient form (4π does not appear) but on the other hand, π appears in the Coulomb law. Conversely, in the Gaussian system, the field equations contain 4π, but the Coulomb law has a simple form.