In practical problems, however, we are given not fab but the incident field F1in.
These two fields are connected via Eqs. (2-12), (2-7) and (2-10) ,
fb Fa +F1 b
fab Fon + 2 arad
4 3C ('nav ) (2-25)3
= Fin 1e i -'b ba) (2-25)
with the help of Eq. (2-11). Substituting this into Eq. (2-24) and using Eqs. (2-16)
and (2-18), one obtains
mnia 2 2a + i2Va) = CbFin, (2-26)
where v2 =- Da ,a. Eq. (2-26) would be equal to the equation of motion derived
from the Lorentz theory of the extended electron by equating the total force on
the electron to zero, if one neglects terms involving higher derivatives of Va the the
second.
To discuss the physical interpretations of Eq. (2-26), one needs to examine
the equation for a = 0 component, describing the energy balance. The right hand
side gives the rate at which the incident field does work on the electron, and is
equated to the sum of the three terms mvo, -2. 7', and -'ii, The first two of
these are the perfect differentials of the quantities mvo and 2.;,, respectively,
and may be considered as intrinsic energies of the electron: the former is the usual
expression for a particle of rest-mass m and the latter the .... 1 I. i. ( i 1-,,,
of the electron [12]. C'!i i ,. in the acceleration energy correspond to a reversible
form of emission or absorption of the field energy near the electron. However, the
third term ( I I,, corresponds to irreversible emission of radiation and gives the
effect of radiation damping on the motion of the electron. According to Eq. (2-
17), this term must be positive since ia is orthogonal to the time-like vector Va
and is thus a space-like vector, and hence its square is negative (in the signature
convention (+1,-1,-1,-1)).