Exchange of helicity in a knotted electromagnetic field

Abstract

There are solutions of Maxwell equations in vacuum in which the magnetic and the electric lines have a nontrivial topology. This behaviour has physical consequences since it is related to classical expressions indicating aspects of the photon content of the electromagnetic field. In this work we present for the first time an exact solution of Maxwell equations in vacuum, having non trivial topology, in which there is an exchange of helicity between the electric and magnetic part of such field. We calculate the temporal evolution of the magnetic and electric helicities, and explain the exchange of helicity making use of the Chern-Simon form. We also have found and explained that, as time goes to infinity, both helicities reach the same value and the exchange between the magnetic and electric part of the field stops.

A class of non-null toroidal electromagnetic fields and its relation to the model of electromagnetic knots

“…For t = 0, equations (5)- (6) are not true. This is due to the fact that the linkage of the field lines may change during time evolution (see [21] for an example, in which this kind of change was presented). However, there is also an important case, called the Rañada-Hopf electromagnetic knot, in which n = m = l = s = 1, where the magnetic and electric helicities take the values given by equations (5)-(6) for any time.…”

On the existence of the field line solutions of the Einstein–Maxwell equations

“…We note that while B i (t, x) has a clear geometric interpretation in terms of magnetic lines, the field E i (t, x) does not and its connection with the electric lines is obscured in the equation (12). Indeed, this less clear geometrical picture is due to the fact that the electric and the magnetic fields are reciprocally orthogonal γ ij E i B j = 0 at every point of U p as can be easily proved.…”

Spin-Orbital Momentum Decomposition and Helicity Exchange in a Set of Non-Null Knotted Electromagnetic Fields

“…We show that the existence of electromagnetic fields in a vacuum with the same constant angular momentum and orbital-spin decomposition, but different electric and magnetic helicities is possible. We find cases where the helicities are constant during the field evolution and cases where they change in time, evolving through a phenomenon of exchanging magnetic and electric components [8]. The angular momentum density presents different time evolution in each case.…”