# L due to Azimuthal (1/r)x(1/r^2) Poynting vector components?

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The basic idea:

I am interested in the possibility of an azimuthally-directed Poynting vector component which drops with the inverse cube of the distance (or as 1/r^3), primarily because it suggests the possibility of emitting field angular momentum, allowing for a uni-directional torque to be exerted on a system with time-varying dipoles.

The question is, "Is this really possible?"

We have the following equations for fields due to an oscillating electric dipole and an oscillating magnetic dipole:

http://www.physicspages.com/2015/01...ed-power-from-an-oscillating-electric-dipole/

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http://www.physicspages.com/2015/01/21/fields-of-an-oscillating-magnetic-dipole/

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In these equations, the r-hat vector points in the direction of varying radial coordinates. We also have a theta-hat vector which is directed along varying polar angle, and phi-hat vector which is directed along varying azimuthal angle. We have some terms that drop with 1/r, some terms that drop with 1/r^2, and yet another that drops with 1/r^3. We also have dependence of these terms based on omega. For a given equation, terms that vary with omega^2 tend to vary with 1/r dependence, while terms that vary with omega^1 tend vary with 1/r^2 dependence. In the case of the magnetic fields of an oscillating magnetic dipole, we also have a term that while dependent on omega (and time) does not scale with omega (or powers thereof), and this term varies with 1/r^3.

One approach to this:

In principle, I can have, at a given point, a radial field component and a polar field component, whose cross product yields an azimuthal component. So in principle, I could take a magnetic field in the r-hat direction and cross it with an electric field in the polar direction to yield a Poynting vector that points in the azimuthal direction.

The equation above for the electric field of a time-varying electric dipole allows for an electric field in the polar direction that drops with the inverse of the distance (as 1/r), while the magnetic field of a time-varying magnetic dipole allows for magnetic field components in the radial direction varying with 1/r^2 or 1/r^3. It seems here that we have the ingredients for a Poynting vector component that is azimuthally-directed and dropping as 1/r^3 (as well as another dropping as 1/r^4). If we take the cross product of the radial vector from the system origin to the sampling point with the Poynting vector at that point, then we get a field angular momentum density that drops with the inverse square of the distance.

Then if integrating this angular momentum density over a hemispherical shell at r (either a +z or -z hemisphere), we would have angular momentum contributions at each r. In principle, the E of the electric dipole and the B of the magnetic dipole could vary such that (one might think) uni-directional angular momentum is "emitted", considering these terms only, and could in principle (one might think) accumulate with time. (One might think) such a system would be capable of emitting angular momentum aligned with the axis of the oscillating dipoles until such a time that the dipoles no longer oscillated. However, the resultant angular momentum is emitted in two opposing directions in the +z and -z hemispheres, such that a net angular momentum is not emitted in this scenario.

A better approach to this:

This could be "fixed" by taking two electric dipoles oscillating on the z-axis but separated (symmetrically) across the origin, while being 180 degrees out of phase. This would produce angular electric and magnetic fields that vary as (1/r^2) resulting in a non-radiating Poynting vector varying as (1/r^4). Furthermore, this would produce an oscillating radial electric field tangent to the xy-plane that drops as 1/r^2 from the origin on the z-axis. Then what is required is to take this (1/r^2) radial electric field and cross it with an (1/r) angular component of the magnetic field to get an azimuthal Poynting vector component that varies as (1/r^3). The z-component of the angular momentum associated with this Poynting vector does point in the same direction in the +z hemisphere as it does in the -z hemisphere, which allows the system to "radiate" angular momentum.

While the angular components of the electric field of the opposed oscillating electric dipoles drop with 1/r^2, the radial components of the magnetic fields drop with 1/r^2 or 1/r^3 which either way is faster than 1/r. Therefore such components do not allow the "emission" of angular momentum that would otherwise alter the contribution described in the previous paragraph.

Also we have radial Poynting vector components due to the angular electric and magnetic field components of the oscillating magnetic dipole. The electric and magnetic field components that drop off with 1/r have an associated radial Poynting vector that drops with 1/r^2. These components would also vary with sin(theta), and the associated Poynting vector would vary with the square of sin(theta), where theta is the polar angle away from the zenith, which leads to a "toroidal" radiation pattern aligned with the dipole axis.

In sum:
Claim #1) The oscillating electric dipoles in opposition would not provide any net radiation due to their net angular fields that vary as (1/r^2) in the xy-plane.
Claim #2) The oscillating magnetic dipole would provide radiation due to its angular fields that vary as (1/r).
Claim #3) The oscillating electric dipoles in opposition, separated by a finite distance from the origin, symmetrically across the origin, provide radial net electric field components relative to the origin that vary as (1/r^2), and coupling this with the (1/r) theta-hat magnetic field components of the oscillating magnetic dipole may result in the "emission" of electromagnetic angular momentum.
The implication is that:
Claim #4) The intensity of the far-field Poynting radiation will depend on the oscillation of the magnetic dipole only.
Claim #5) The intensity of the "emitted" electromagnetic angular momentum will depend on the oscillation of the magnetic dipole as well as the oscillation of the opposed electric dipoles.
Thus:
Claim #6) The intensity of the "emitted" electromagnetic angular momentum may be "decoupled" and increased relative to the intensity of the emitted electromagnetic energy without reducing the angular frequency (i.e. by choosing a more intense pair of electric dipole moments to oscillate in opposition).

Another note:
Claim #7) Just as the intensity of the emitted radiation would increase with the fourth power of the angular frequency (omega), the intensity of the "emitted" angular momentum would increase with the fourth power of the angular frequency (omega), since the electric fields and magnetic fields of all the oscillating dipoles have components that vary with the square of the angular frequency (omega) which contribute to the "emitted" angular momentum.

Reality check:
According to Planck–Einstein relation, the relationship between energy (E) and the angular momentum (h-bar) of a photon is determined by its angular frequency (omega). So can claim #6 really be possible or did I make a mistake? If it is a mistake, where did I go wrong?

Sincerely,

Kevin M.

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