Covariance of the Abraham-Lorentz Formula for Radiated Power, Charge Oscillating Along the y-axis
G.R.Dixon, 7/4/2006
The relativistic Abraham-Lorentz formula for the power radiated by a charge moving periodically along the y-axis of inertial frame K is
. (1)
For purposes of numerically integrating PRad(t), we again build a table with columns:
|
ti |
uy(ti) |
g P(ti) |
ay(ti) |
day/dt(ti) |
PRad(ti) |
If the charge’s motion is
, (2)
then consecutive values of ti can be separated by increments dt, where
. (3)
Having populated the table, the energy radiated per cycle is:
. (4)
For q=1E-6 coul, A=1E-6 meters,
w=.0001c/A, Eq. 4 yields
, (5)
which is the same result obtained using the Larmor-Lienard formula.
Given the motion in Eq. 2, q radiates a spherical wave. In frame K the net momentum in one complete wave is zero. Thus from the perspective of inertial frame K’ (moving in the positive x-direction of K at speed v),
. (6)
We can determine whether Abraham-Lorentz (Eq. 1) produces this result in K’ by transforming the quantities in the table into K’ quantities.
Fig. 1a plots P(t), and Fig. 1b plots P’(t’), with v=.95c. Note that, for this oscillation along the y-axis, the K and K’ plots have the same shape, differing only in the scales on the time axes. (When motion was along the x-axis, the shapes were distinctly different for relativistic values of v.) Also noteworthy in frame K is the phase shift of
p/2 from the Larmor-Lienard plot. When applied to the K’ values, the sum in Eq. 4 produces
. (7)
(Larmor-Lienard produces 60312 joules.) Also,
. (8)
Figure 1a
P(t)
Figure 1b
P’(t’)