Abdullah Eroglu - RF/Microwave Engineering and Applications in Energy Systems

(1.92)

Since , then

(1.93)

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.

(1.96)

(1.97)

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)

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

(1.103)

We now apply Stokes' theorem as described in Section 1.3.6for the left‐hand side of the equation in (1.103)as

(1.104)

From (1.103)and (1.105), we can now write as

(1.105)

or

(1.106)

In (1.106) i sis the current through the surface S . Equation (1.106)is the integral form of the equation given in (1.89).

Similarly, we can express the integral form of Eq. (1.88)as

(1.107)

As a note, it is assumed that the magnetic current source does not exist. Taking the volume integral of both sides over volume V and surface S gives

(1.108)

We now apply divergence theorem as described in Section 1.3.5for the left‐hand side of the equation in (1.108)as

(1.109)

From (1.108)and (1.109), we can write

(1.110)

Equation (1.110)is the integral form of the equation given in (1.91). Similarly, the integral form of Eq. (1.90)can be found as

(1.111)

In summary, the integral forms of Maxwell's equations are

(1.112)

(1.113)

(1.114)

(1.115)

Let's assume we have a sinusoidal function that changes in position and time. This function can be expressed as

(1.116)

Equation (1.116)can rewritten as

(1.117)