Maxwell equation for static fields.explain how they are modified for time varying electric and magnetic fields
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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 intensityE 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/m3]
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)
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 intensityE 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/m3]
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)
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