Antennas. Yi Huang

      where

       ρ is the charge density.

        is a vector operator;

       ∇× is the curl operator or called rot in some countries instead of curl.

       ∇• is the divergence operator.

      Here we have both the vector cross product and the dot product.

Figure 1.13 James Clerk Maxwell

Maxwell’s equations describe the interrelationship between electric fields, magnetic fields, electric charge, and electric current. Although Maxwell himself was not the originator of the individual equations, he derived them again independently in conjunction with his molecular vortex model of Faraday's lines of force, and he was the person who first grouped these equations all together into a coherent set. Most importantly, he introduced an extra term to Ampere’s Circuital Law, the second equation of (1.19). This extra term is the time derivative of the electric field and is known as Maxwell's displacement current. Maxwell's modified version of Ampere's Circuital Law enables the set of equations to be combined together to derive the EM wave equation, which will be further discussed in Chapter 3.

      Now let us have a closer look at these mathematical equations to see what they really mean in terms of the physical explanations.

      1.4.3.1 Faraday's Law of Induction

      (1.30)

      This equation simply means that the induced electromotive force (EMF with a unit in V, it is the left‐hand size of the equation expressed in the integral form as shown in Equation (1.37)) is proportional to the rate of change of the magnetic flux. In layman's terms, moving a conductor (such as a metal wire) through a magnetic field produces a voltage. The resulting voltage is directly proportional to the speed of movement. It is apparent from this equation that a time‐varying magnetic field will generate an electric field, i.e. E ≠ 0. But if the magnetic field is not time varying, it will NOT generate an electric field!

      1.4.3.2 Amperes’ Circuital Law

      (1.31)

This equation was modified by Maxwell by introducing the displacement current . It means that a magnetic field appears during the charge or discharge of a capacitor. With this concept, and the Faraday’s law, Maxwell was able to derive the wave equations, and by showing that the predicted wave velocity was the same as the measured velocity of light, Maxwell asserted that light waves are EM waves.

      This equation shows that both the current (J) and time‐varying electric field can generate a magnetic field, i.e. H ≠ 0.

      1.4.3.3 Gauss' Law for Electric Field

      (1.32)

      This is the electrostatic application of Gauss's generalized theorem, giving the equivalence relation between any flux, e.g. of liquids, electric or gravitational, flowing out of any closed surface and the result of inner sources and sinks, such as electric charges or masses enclosed within the closed surface. As a result, it is not possible for electric fields to form a closed loop. Since D = εE, it is also clear that charges (ρ) can generate electric fields, i.e. E ≠ 0.

      1.4.3.4 Gauss’ Law for Magnetic Field

      (1.33)

      This shows that the divergence of the magnetic field (∇ • B) is always zero, which means that the magnetic field lines are closed loops, thus the integral of B over a closed surface is zero.

      For a time‐harmonic EM field (which means the field linked to the time by factor e^jωt where ω is the angular frequency and t is the time), we can use the constitutive relations

      (1.34)

      to write Maxwell’s equations into the following forms

(1.35)

      where B and D are replaced by the electric field E and magnetic field H to simplify the equations and they will not appear again unless necessary.

It should be pointed out that, in Equation (1.35), can be viewed as a complex permittivity defined by Equation (1.20). In this case, the loss tangent is

      (1.36)

      It is hard to predict how the loss tangent changes with the frequency since both the permittivity and conductivity are functions of frequency as well. More discussion will be given in Chapter 3.

      1.4.4 Boundary Conditions

      Maxwell’s equations can also be written in the integral form as

(1.37)

Consider the boundary between two materials shown in Figure 1.14. Using these equations, we can obtain a number of useful results. For example, if we apply the first equation of Maxwell’s equations in integral form to the boundary between Medium 1 and Medium 2, it is not difficult to obtain [2]:

      (1.38)

      where is the surface unit vector from Medium 2 to Medium 1 as shown in Figure 1.14. This condition means that the tangential components of an electric field () are continuous across the boundary between any two media.