Arbitrary current vector

Consider next the current operator ju(x). The correspondence principle suggests that its form is j ( ) = — (e/2)( ( )yB, (a )]. Such a form foijn(x) does not satisfy Eq. (11-477). In fact, due to covariance, the spectral representation of the vacuum expectation value of 8u(x)Av(x), where ( ) is an arbitrary four-vector, is given by... 

In Eq. (61), the shift of origin is represented by an arbitrary constant vector d. Within the CTOCD-PZ approach, a general transformation function d — d(r) is sought, specifying the origin of the coordinate system in which the paramagnetic contributions to the current density is formally annihilated. This function is evaluated point-wise via the condition determined by Eq. (67). The IJi.s. of this relationship vanishes for... 

We can now construct the total fields due to an arbitrary current distribution within the fiber. For an individual tube, the electric and magnetic fields are linearly related to the vector potential. Thus the relationships corresponding to Eq. (34-24) are... 

Another interesting property of the current density is that the total electronic energy of an arbitrary system in the presence of any external field whatever, with vector potential A, may be written [5]... 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is... 

The path L can have an arbitrary shape, and it can intersect media characterized by various physical properties. In particular, it can be completely contained within a conducting medium. Because of the fact that the electromotive force caused by electric charges is zero, a Coulomb force field can cause an electric current by itself. This is the reason why non-Coulomb forces must be considered in order to understand the creation of current flow. Equation 1.46 is the first Maxwell equation for electric fields which do not vary with time, given in its integral form, and relates the values of the field various points in the medium. To obtain eq. 1.46 in differential form, we will make use of Stoke s theorem, according to which for any vector A having first spatial derivatives, the following relationship holds ... 

First of all we will present a vector potential at an arbitrary point F as a sum of vector potentials caused by all elements of the current ring (Fig. 4.42b). The vector potential of the current element, dl, at the point P is written as ... 

Werner, D.H. 1993. An exact formulation for the vector potential of a cylindrical antenna with uniformly distributed current and arbitrary radius. IEEE Trans, on Antennas and Propagation 41 (8) 1009-... 

When a current source is located within a clad fiber of arbitrary profile, the determination of the radiation field is extremely complicated. However, if the fiber is weakly guiding the determination is greatly simplified [2]. It is intuitive that when the variation in the refractive-index profile is small, the source radiates as if it were located in a virtually uniform, infinite medium of refractive index equal to the cladding index n j. The problem is then analogous to the radiation from an antenna in free space. Consequently, we can borrow from standard antenna theory, and couch the solution to radiation from the weakly guiding fiber in terms of the electromagnetic vector potential A [3-5]. 

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