We have examined Maxwell's equations in differential and integral forms. Some of the advantages of eliminating the differential dependence of the field quantities on one or more of the space and time variables by means of Fourier (or Laplace) transformations have been discussed. In particular, similarities and differences between frequency-domain analysis and phasor-domain analysis have been highlighted. Duality between electric and magnetic quantities was pointed out.
The problem of relating field vectors to their sources in free space has been solved in two equivalent ways. Firstly, we have expressed the electric and magnetic fields at any point in space-time as volume integrals over their sources. Secondly, we have introduced scalar and vector potentials, from which electric and magnetic fields are easily obtained by differentiation, and have expressed each of these potentials as a volume integral over a source term. The advantages of introducing scalar and vector potentials, as well as higher-order potentials, have been presented.
When a material medium is present, a comparison between a macroscopic approach and a microscopic one allows for the utilization of the results obtained in free space; the electric and magnetic fields are expressed as integrals over the volume of equivalent sources, which comprise the true charges and current, plus equivalent charges and currents arising from the polarization and magnetization vectors. Once the constitutive relations are specified, one is left with two coupled integro-differential equations for the electric and magnetic fields.
In the above treatment, the existence of electric and magnetic fields at any point inside a material medium is postulated, without considering the problem of their measurements (Maxwell's approach). From an experimental viewpoint, if a measuring probe is inserted in a medium, a cavity is created in which the measurement takes place (Faraday's approach). The relationship between the fields inside the cavity and the Maxwellian field is conceptually important, and is considered at the end of the next chapter.