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Flow of Radiant T.E. Field Energy

Any loop antenna radiates only for a finite amount of time. Consequently, one can calculate the flow of total emitted energy, which is given by the spatial integral, Eq.(57). The fact that is a cyclic coordinate for axially symmetric sources and radiation implies that for the T.E. field. Consequently, the spatial integral, a conserved quantity independent of time , reduces with the help of the table of derivatives in Section IV to

 (65) (66)

The computation leading to the last line has been consigned to the Appendix. This computed quantity is the total energy flow (energy velocity), or equivalently, the total momentum into the -direction, radiated by a magnetic dipole accelerated uniformly in Rindler sector (upper sign) or in Rindler sector (lower sign).

 (67)

with

is a linear functional correspondence. It maps the temporal history at in Rindler sector (resp. ) with 100% fidelity onto a readily measurable e.m. field along the -axis (or a line parallel to it) on the spatial hypersurface in Rindler sector . This correspondence has 100% fidelity because, aside from a -independent factor, the source history and the scalar field differ only by a constant -independent shift on their respective domains and . This implies that
 (68)

when applied to the source function . The expression for the radiated momentum becomes more transparent physically if one uses the proper time derivative

at the source. Introduce the (proper) magnetic moment of the current loop having radius :

One finds from Eq.(66) that the proper radiated longitudinal momentum (i.e. physical, a.k.a. orthonormal, component of energy flow pointing into the -direction) measured per proper spatial -interval in is
 (69)

This is the formula for the proper radiant energy flow due to a magnetic dipole moment subject to uniform linear acceleration . There are two factors of . The the first converts the coordinate -momentum component into its physical component. The second is due to the fact that Eq.(69) expresses this quantity per proper distance into the direction.

When the acceleration is small then the second term becomes small compared to the first. In fact, one recovers the familiar Larmor formula [#!Landau1962!#] relative to a static inertial frame by letting and letting , which corresponds to inertial motion. By contrast, Eq.(69) is the correct formula for an accelerated dipole moment. However, observation of radiation from such a source entails that the measurements be made relative to an expanding inertial reference frame.

Next: Flow of Radiant T.M. Up: RADIATED POWER Previous: Electric Dipole and its
Ulrich Gerlach 2001-10-09