Anand K. Verma - Introduction To Modern Planar Transmission Lines

The set of Maxwell’s equations, given below, consists of four time‐dependent vector equations; relating the sources such as conduction current density ( ), electric displacement current density , and magnetic displacement current density to the magnetic field ( and the electric field ( respectively:

(4.4.1)

All quantities in the above equations are space‐time dependent. The current densities , , and are not the power supplying sources to the propagating EM‐wave in a medium. These current densities are created by the externally applied magnetic and electric current densities and supplying power to the EM‐wave and partly getting absorbed as loss in a conducting medium.

The right‐hand sides of the above equations could be treated as the sources ( excitations ) and the left‐hand fields as the responses . The force field quantities ( , ), flux field quantities ( ), and current densities are functions of both the space variables and time variable. Further, in any material medium, the flux field quantities are related to the force field quantities by the constitutive relations, given in equation (4.1.7). The conduction current density in a lossy medium is related to the electric field, as given in equation (4.1.9).

In general, ε r, μ r, σ are the tensors quantities for an anisotropic medium. However, these are scalar quantities for an isotropic medium. They are also treated as complex quantities to include the losses of a medium. In the case of a complex conductivity σ *, its real part is responsible for the loss in a medium, whereas its imaginary part accounts for the energy storage. In a dispersive medium, ε r, μ rand σ are also frequency‐dependent. The characteristics of various kinds of media, such as dielectrics, conductors, plasma, semiconductors, ferrites, and so forth are accounted for in Maxwell’s equations through the constitutive relations applicable to these physical media. Maxwell’s equations, along with the constitutive relations, are the field equations, not the force equations, i.e. these equations do not express the forces exerted by the fields on stationary or moving charges. This is achieved through Lorentz’s force equation :

(4.4.2)

where q is the charge on a mass m that is moving with velocity v. In this case, Lorentz's force acting on a moving charge is equal to Newtonian force:

(4.4.3)

Lorentz's force has two components: (i) electric force component and (ii) magnetic force component . The electric force is exerted either on a moving or on a stationary charge in the static, or in the time‐varying electric field. The magnetic force is exerted only on a moving charge in the static, or in the time‐varying, magnetic field. In the case of a time‐varying electric, or the time‐varying magnetic field, both fields are always present and are related through Maxwell’s equations (4.4.1a) and (4.4.1b). Thus, both components of Lorentz's force are present on a moving charge in a time‐varying EM‐field.

It is observed that in the absence of the external sources, in a lossless medium Maxwell’s equation (4.4.1a)states that a time‐varying magnetic field creates a time‐varying electric field ; and Maxwell’s equation (4.4.1b)states that a time‐varying electric field ( ) creates a time‐varying magnetic field . Thus, Maxwell’s equations (4.4.1a)and (4.4.1b)form a set of the coupled equations, showing an interdependence of the time‐varying electric and magnetic fields. It is like the two‐variable simultaneous equations that occur in ordinary algebra. However, in the present case, the field variables ( , are vector quantities. The coupled partial differential equations are solved either for the or by following the rules of the vector algebra. The solutions provide the wave equation either for the electric ( ) or for the magnetic ( ) field.