International Journal of Magnetics and Electromagnetism
Volume 6, Issue 1
Angular Momentum Emission by a Rotating Dipole
Radi I Khrapko*
Table of Content
Figure 1: Polarization of the electric field seen....
Polarization of the electric field seen by looking form different directions at a circular oscillator.
Figure 2: Torque distribution.....
Figure 3: Spin flux distribution....
Spin flux distribution.
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Radi I Khrapko*
Moscow Aviation Institute, Volokolamskoe Shosse 4, Russia
Radi I Khrapko, Moscow Aviation Institute, Volokolamskoe Shosse 4, 125993, Moscow, Russia.
Accepted: December 09, 2020 | Published Online: December 11, 2020
Citation: Khrapko RI (2020) Angular Momentum Emission by a Rotating Dipole. Int J Magnetics Electromagnetism 6:026.
Copyright: © 2020 Khrapko RI. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
A new calculation confirms the presence of spin radiation along the axis of rotation of a dipole. This is further proof of the need to introduce the spin tensor into classical electrodynamics, along with the energy-momentum tensor.
Classical spin, Electrodynamics, Spin radiation
JH Poynting : "If we put E for the energy in unit volume and G for the torque per unit area, we have ".
This means that such radiation is Weyssenhoff's spin-fluid .
J Weyssenhoff: "By spin-fluid we mean a fluid each element of which possesses besides energy and linear momentum also a certain amount of angular momentum, proportional - just as energy and the linear momentum - to the volume of the element".
Where is the free electromagnetic field Lagrangian, is the vector potential, and is the field-strength tensor. The local sense of a spin tensor is as follows. [J*s/m3] is spin volume density, [J*s/m2] is spin flux density, i.e. torque per unit area (cf. J. H. Poynting). The spin tensor is used in the publications [9-20]. However, the spin tensor is ignored in works expressing the common point of view, e.g. [21-25].
The local sense of the energy-momentum tensor is as follows. [N*s/m3] is momentum volume density, [kg/m2*s] is mass-energy flux density. It means, e.g., is the momentum in the volume .
Moment of momentum, e.g., is the orbital angular momentum of the momentum contained in the volume . So, the total angular momentum possessed by the volume is
The total torque per the area , i.e. angular momentum flux, is
It is important that spin is not associated with a moment of a linear momentum, or even with a motion of matter. Hehl writes about spin of an electron :
"The current density in Dirac's theory can be split into a convective part and a polarization part. The polarization part is determined by the spin distribution of the electron field. It should lead to no energy flux in the rest system of the electron because the genuine spin 'motion' take place only within a region of the order of the Compton wavelength of the electron".
Electromagnetic Field of a Rotating Dipole
The first terms of (5), (6) are proportional to and so represent radiation. This radiation is of circular polarization in the direction of the rotational axis, z-axis (see Figure 1 from ). Therefore this field contains the spin flux . We calculate this spin flux per sphere in Section 3.
At the same time this radiation contains no orbital angular momentum flux per elements of the sphere . . Really, the first terms fields E & H are orthogonal to each other and to the vector r. So, in any point, we can enter local Cartesian coordinates such that , , i.e. are not equal to zero only. Using this coordinates we find according to (2): . So the orbital angular momentum is not radiated.
The second terms field of (5), (6) contains the orbital angular momentum flux, or torque, per the sphere . In Refs [32-37], spherical coordinates were used, and the angular distribution of the torque was obtained (see Figure 2):
Where, . This torque is located in the neighborhood of the plane of rotation where the polarization is near linear. This torque is not radiated. This torque is like a static torque that someone can apply (Figure 2).
Spin Radiation by a Rotating Dipole
Spin radiated by the first terms field was calculated in  using the spin volume density on the assumption that this density is moving at the speed of light. Here the spin flux density is used. This is more naturally.
Accordingly to , we have
Because of , we need the Cartesian coordinates of elements of the sphere , which spherical coordinates are . The transformation coefficients are;
, and . So we have
This result, , is coincided with Ref. . The angular distribution of the spin radiation is represent in Figure 3.
A rotating electric dipole emits angular momentum flux of two types: (i) Spin flux, which is directed mainly along the axis of rotation and determined by the spin tensor, and (ii) Orbital angular momentum flux determined by the energy-momentum tensor. The spin flux is not recognized by nowadays electrodynamics.
I am eternally grateful to Professor Robert Romer for the courageous publication of my question: "Does a plane wave really not carry spin?"  (was submitted on 07 October, 1999).