Anand K. Verma - Introduction To Modern Planar Transmission Lines

Equation (4.4.1c)of Maxwell's equations is a divergence relation. It shows that the electric field originates from a charge, and ends on another charge. Equation (4.4.1d)shows that the divergence of the magnetic field is zero; meaning thereby that a magnetic charge does not exist in nature. The magnetic field exists as a closed‐loop around a current‐carrying conductor. However, sometimes a presence of the hypothetical magnetic charge is assumed, and the divergence equation (4.4.1d)is modified as . This assumption maintains the symmetry of Maxwell’s equations. Likewise, to maintain the symmetry of Maxwell’s equations, a hypothetical magnetic current density (−J m) term is also to be added to equation (4.4.1a). The modified Maxwell equations in the symmetrical form are given below:

(4.4.4)

Figure 4.8 Surfaces and volume used in the integral form of Maxwell equations.

In a real material medium, a current can flow due to the time‐dependent electric polarization (P), and also due to the time‐dependent magnetic polarization (M). These are known as the electric and magnetic polarization currents and are incorporated in Maxwell’s equations as the electric and magnetic displacement current densities. Sometimes, the external sources, magnetic current density and electric current density are further added to Maxwell equations (4.4.1a) and (4.4.1b). The external current sources are retained in the modified form of Maxwell equations. Figure (4.8a)shows externally impressed current sources, creating fields in an enclosure. These externally applied current sources supply power to create the electric field and magnetic field in a material medium. They also cause the flow of current in a lossy medium. It is examined in the next section while discussing the energy balance of the electromagnetic field [B.8, B.9]. The lossless free space is treated as a charge‐free and source‐free medium, i.e. .

A divergence of Maxwell’s equations (4.4.1b)in absence of and using equation (4.4.1d), leads to the following continuity equation :

(4.4.5)

The above equation shows the conservation of charge. In any enclosed volume, the decrease of charge is associated with a flow of current out of the volume.

Normally, the time‐harmonic fields are used in most of the applications. The time‐dependent electric field and other field quantities are written in the phasor form for the time‐harmonic field:

(4.4.6)

(4.4.7)

It is noted that the same notation is used for both the time‐space dependent fields [ and only space‐dependent fields [ in the phasor form. The context of the discussion can clarify the situation. In general, the space coordinates based [ etc.] phasor fields are complex quantities, with both the magnitude and phase angle associated with them. The real part gives actual sinusoidal field quantity with phase relation. is the instantaneous field quantity without any phase term. Further, the following relations are useful:

(4.4.8)

Using the above equation for the electric field and similar equation for the magnetic field, Maxwell’s equations (4.4.1), in the external source free medium, are rewritten below in the phasor form for the time‐harmonic field:

(4.4.9)

where flux density vectors are related to field intensity vectors in an anisotropic medium by the following tensor constitutive relations:

(4.4.10)

In the case of an external source‐free homogeneous isotropic medium, Maxwell’s equations are written in terms of field intensities only:

(4.4.11)

where the scalar constitutive relations are , , . The differential form of the above given Maxwell’s equations provide relations between the electric field and the magnetic field at any location in the medium. Also, the sources are specified at a point in the medium. The Maxwell equations (4.4.9) involving the flux densities and field intensities apply to both the isotropic and anisotropic media, whereas Maxwell’s equations (4.4.11) involving the field intensities apply to the isotropic medium only .