Electromagnetic field equations

In previous sections, by making use of Gauss s and Stoke s theorems, we have developed the basic laws for the electromagnetic fields in the form of equations. In accord with these laws the electromagnetic field must satisfy the following set of equations  

The first equation (eq. 1.246) is in essence Faraday s law, while the second equation (eq. 1.247) follows from a combination of Ampere s law and the postulate of conservation of charge. The third equation (eq. 1.248) is obtained from Coulomb s law for a nonalternating electric field. However, it remains valid regardless of how quickly the field may change. In order to demonstrate this we will use the postulate of conservation of charge (eqs. 1.131-1.132)  

Applying the second Maxwell equation (1.247), twice along contour L, once in one 

Adding the two equations and considering that the surfaces Si and S2 form a closed surface, we obtain  

By analogy, using the first Maxwell equation, we also have BdS = 0 

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In a first step, these conditions can be imposed on the general solutions of the present electromagnetic field equations. At a later stage the same equations... 

The present theory has been developed in terms of an extended Lorentz invariant form of the electromagnetic field equations, in combination with an addendum of necessary basic quantum conditions. From the results of such a simplified approach, theoretical models have been obtained for a number of physical systems. These models could thus provide some hints and first... 

The electromagnetic field equations on the 0(3) level can be obtained from this purely geometrical theory by using Eq. (631) in the Bianchi identity... 

In Section V it will be shown that the quaternion structure of the fields that correspond to the electromagnetic field tensor and its current density source, implies a very important consequence for electromagnetism. It is that the local limit of the time component of the four-current density yields a derived normalization. The latter is the condition that was imposed (originally by Max Bom) to interpret quantum mechanics as a probability calculus. Here, it is a derived result that is an asymptotic feature (in the flat spacetime limit) of a field theory that may not generally be interpreted in terms of probabilities. Thus, the derivation of the electromagnetic field equations in general relativity reveals, as a bonus, a natural normalization condition that is conventionally imposed in quantum mechanics. 

Let us now focus on the irreducible expressions of the electromagnetic field equations in special relativity, using the quaternion calculus. We will then come to their form in general relativity. 

This result, according to Eq. (49), in turn, implies that Eq. (52), F[p> 7] = 0, must be true, indicating that there are no magnetic monopoles in this formulation of the electromagnetic field equations—for if there were, there would be a nonzero source term in Eq. (52). 

G. Mur, Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations, IEEE Trans. Electromagn. Compat, vol. 23, no. 4, pp. 377-382, Nov. 1981. 

For further applications we have to write down the electromagnetic field equations in 4-dimensional form. Then we have to use the Lorentz gauge. The Eqs(34) for the potentials become ... 

G. Kron, Equivalent circuits to represent the electromagnetic field equations, Phys. Rev., 64, 1943, 126-128. 

Maxwell, who was born in 1831, the year that Faraday published his discovery of electromagnetic induction, expressed Faraday s discoveries mathematically in field theory in 1855-1857, followed by his major work in four electromagnetic field equations, expressed in vector form as follows ... 

These allow us to condense the electromagnetic field equations into a single vector equation with E and n, which appears through e, as... 

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