is derived by taking the divergence of Eq. (1.89) and using Eq. (1.91) as
(1.92)
Since
Equation (1.93) also represents the fundamental law of physics which is known as conservation of an electric charge.
In an isotropic medium, the material properties do not depend on the direction of the field vectors. In other words, the electric field vector is in parallel with the electric flux density and the magnetic field vector is in parallel with the magnetic flux density.
(1.94)
(1.95)
ε is the permittivity of the medium and represents its electrical properties and μ is the permeability of the medium and represents its magnetic properties. They are both scalar in the existence of an isotropic medium. However, when the medium is anisotropic this is no longer the case. The electrical and magnetic properties of the medium depend on the direction of the field vectors. Electric and magnetic field vectors are not in parallel with electric and magnetic flux. So, the constitutive relations get the following forms for an anisotropic medium.
The permittivity and permeability of anisotropic medium are now tensors. They are expressed as
(1.98)
and
(1.99)
Then, (1.96) and (1.97) take the following form in a rectangular coordinate system.
(1.100)
(1.101)
(1.102)
1.4.2 Integral Forms of Maxwell's Equations
Let's begin our analysis with Eq. (1.89). Taking the surface integral of both sides of Eq. (1.89) over surface S with contour C gives
We now apply Stokes' theorem as described in Section 1.3.6 for the left‐hand side of the equation in (1.103) as
(1.104)
