Force lines

We have seen that the field dispersion can not occur only if a reluctance variation of the medium in which forces lines are oriented exists. 

Stressing the need to paint certain states of mind stati d animo)," uniquely perceived hy a modernist clairvoyant artist, and to depict those wholly invisible force-lines (linee della forza)" stirring the vitalist universe so beloved of the Occultists, the apocalyptic Futurist message concludes (with italic emphasis), We have proclaimed ourselves to be the primitives of a completely renovated sensitiveness. ... 

The [Futurist] Exhibitors to the Public (1912), in Chipp, 294-98 for the argument that those states of mind (stati d animo) and force-lines (linee della forza) were taken direkt vom Okkultismus, see the essays dissecting Futurist notions collected in Loers, 431-91. 

Figure 2.5 The orbital and spin motions of an electron around a nucleus are equivalent to circular electric currents I and i these produce magnetic fields shown by their force lines (broken curves)...

The physical mechanism is this inhomogeneity of the velocity field leads to a flow which exponentially stretches the force lines of the initial small magnetic field. The growth continues until the forces generated by the magnetic field become equal to the hydrodynamic forces and begin to influence the flow. 

Along with the dissipation term, rV2h, this equation contains the term (h- V)v which describes the growth of the magnetic field with the stretching of the magnetic force lines noted by Batchelor. Thus, we cannot assert that h or h2 is eliminated and that J h2 dV decreases monotonically. 

Fig. 1.1. A magnetic moment can be seen as a magnetic dipole n characterized by north (N) and south (S) polarities. It gives rise to a magnetic field which is indicated by force lines. The dipolar nature provides the vectorial nature of this moment, whose intensity is indicated by H-...

In all our experiments the two magnetic bars are immersed in an external magnetic field. The intensity of the magnetic field is proportional to the density of force lines (Fig. 1.4). Later, we will be interested in the effective field in a given region of space, which is referred to as magnetic induction B (expressed in tesla) ... 

Fig. 1.4. The force lines of a magnetic field Bn. (A) A weak, homogeneous field. (B) A strong, homogeneous field. (C) An inhomogeneous field with weak and strong field regions.

A. Force Lines, Vortex Atoms, Topology, and Physics... 

C. Faraday s Conception of Force Lines Suggests a Topological Structure for Electromagneti sm... 

It is based on the idea of electromagnetic knot, introduced in 1990 [27-29] and developed later [30-32], An electromagnetic knot is defined as a standard electromagnetic field with the property that any pair of its magnetic lines, or any pair of its electric lines, is a link with linking number i (which is a measure of the extent to which the force lines curl themselves around one another, i.e., of the helicity of the field). These lines coincide with the level curves of a pair of complex scalar fields , 0. The physical space and the complex plane are compactified to Si and S2, so that the scalars can be... 

The topology of the force lines thus induces a topological structure in the set of the fields of the model. 

E = V x C. This implies an interesting interpretation of the Hopf index n, since that helicity is equal to the classical expression of the difference between the numbers of right-handed and left-handed photons contained in the field Nr — Nr (defined by substituting Fourier transform functions for creation and annihilation operators in the quantum expression). In other words, n = Nr — Nr- This establishes a relation between the wave and the particle understanding of the idea of helicity, that is, between the curling of the force lines to one another and the difference between right- and left-handed photons contained in the field. 

To summarize this subsection, the description of the dynamics of the force lines as the level curves of two maps. S 3i -rS2, given by two complex functions topological structure, in such a way that the mere existence of a pair of such functions guarantee that the corresponding pullbacks of the area 2-form in S2 automatically obey the Maxwell s equations in empty space. 

Figure 1. Schematic aspect of several force lines (either magnetic or electric) of an electromagnetic knot. Any two of the six lines shown are linked once.

This equation shows a close relation between the wave and particle aspects of the helicity. On the left side, the wave helicity is the semisum of the electric and magnetic helicities that characterizes the topology of the force lines as a function of the linking number of the pairs of electric lines and of the magnetic lines. On... 

Note that (139) are highly nonlinear in the scalars but become exactly the linear Maxwell equations in the fields F v and F 1V. In this sense, the Maxwell equations are the exact linearization (by change of variables, not by truncation ) of a nonlinear theory with topological properties, in which the force lines... 

One of these topological constants of the motion is the electromagnetic helicity, defined as the semisum of the magnetic and electric helicities, which is equal to the linking number of the force lines... 

---