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Theory
Introduction to Electromagnetics Maxwell's Equations Near-Field and Far-Field Transmission Line Models Scale modeling References Symbols and units Constants Mutual Impedance Between Wire Elements
2 Maxwell’s Equations
James Clerk Maxwell (1837-1879) gathered all prior knowledge in electromagnetics and summoned the whole theory of electromagnetics in four equations, called the Maxwell’s equations. To evolve the Maxwell’s equations we start with the fundamental postulates of electrostatics and magnetostatics. These fundamental relations are considered laws of nature from which we can build the whole electromagnetic theory. According to Helmholtz’s theorem, a vector field is determined to within an additive constant if both its divergence and its curl are specified everywhere [ 8 ]. From this an electrostatic model and a magnetostatic model are derived only by defining two fundamental vectors, the electric field intensity E and the magnetic flux density B , and then specifying their divergence and their curls as postulates. Written in their differential form we have for the electrostatic model the following two relations' [ 8 ]: Equation 6 Equation 7 where r is the volume charge density: [C/m^3 ] Equation 8
These are based on the electric field intensity vector, E , as the only fundamental field quantity in free space. Then to account for the effect of polarization in a medium the electric flux density, D , is defined by the constitutive relation: Equation 9 where the permittivity e is a scalar (if the medium is linear and isotropic). Similarly for the magnetostatic model we have the following two relations, based on the magnetic flux density vector, B , as the fundamental field quantity: Equation 10 Equation 11 where J is the current density. To account for the material here as well, we define another fundamental field quantity, the magnetic field intensity, H , and we get the following constitutive relation: Equation 12 where m is the permeability of the medium. Using the constitutive relations we can rewrite the postulates and the relations derived is gathered in the following table: Table 1 Fundamental Relations for Electrostatic and Magnetostatic Models (The Governing Equations)
The Governing Equations
Electrostatic Model Equation 13 Equation 14 Magnetostatic Model Equation 15 Equation 16 These equations must, however, be revised for calculation of time varying fields. The electrostatic model must be modified due to the observed fact that a time varying magnetic field gives rise to an electric field and vice versa and the magnetostatic model must be modified in order to be consistent with the equation of continuity. The complete model for electromagnetic fields (Maxwell’s equations) is gathered in the following table (Table 2), where the integral forms of the equations are added [ 8 ]: Table 2 Maxwell's Equations, both in differential and integral form
Equation 26 where: is the wave number r is the distance between the radiating source and the point of observation Using the constitutive relations, Equation 9 and Equation 12, the magnetic field intensity H can then be calculated as: Equation 27 The electric field intensity E can also be calculated using potential functions by introduction of the scalar electric potential V : Equation 28 When both the vector magnetic potential A and the scalar electric potential V are known, the electric field intensity E is derived by: Equation 29 It is however not necessary to calculate both the magnetic field intensity H and the electric field intensity E since they are related by the equation: Equation 30 Previous: Introduction to Electromagnetics | Next: Near-Field and Far-Field EMC of Telecommunication Lines A Master Thesis from the Fieldbusters © 1997 Joachim Johansson and Urban Lundgren
