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Covariance of the Larmor-Lienard Formula for Radiated Power, Charge Oscillating Along y-axis

G.R.Dixon, 7/06/2006

Suggested reading: Covariance of the Larmor-Lienard Formula for Radiated Power, Oscillation Along the x-axis … Discusses the case when the oscillation is along the x-axis.

The Larmor-Lienard formula for the power radiated by a charge moving along the y-axis of inertial frame K is

. (1)

For purposes of numerical integration of PRad(t), we can build a table with the following columns:

ti

uy(ti)

gP(ti)

ay(ti)

PRad(ti)

If the charge’s motion is periodic, say

, (2)

then consecutive values of ti can be separated by increments dt, where

. (3)

Having populated the table, the energy radiated per cycle can be computed:

. (4)

For q=1E-6 coul, A=1E-6 meters, w=.0001c/A, Eq. 4 yields

. (5)

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 Larmor-Lienard (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 the identical shapes, but the different scales on the time axes. When applied to the K’ values, the sum in Eq. 4 produces

, (7)

and

. (8)

Figure 1a

                                                                   

P(t)

Figure 1b

                                                                    

P’(t’)