This is all one can get from the laws of conservation of energy and momentum.
To develop this further into the equation of motion for the electron, one needs to
fix the vector Ba by making some assumptions. Taking a dot product of the both
sides of Eq. (2-20) with v', we have
VaBa -= e2- vaa -(, *', fab 0, (2-21)
2
by Eq. (2-17) and from the antisymmetry of the tensor fab. Then we may assume
that Ba could be any vector function of Va and its derivatives. The simplest choice
that satisfies Eq. (2-21) would be
Ba = k,, (2-22)
where k is a constant.
Substituting Eq. (2-22) into the right hand side of Eq. (2-20), one sees that the
constant k must be of the form
k = t2-1 m, (2-23)
where m is another constant independent of e, in order that our equations may
have a definite limiting form when c tends to zero. Then one gets
mia = eVbfab, (2-24)
as the equations of motion for the electron. This is the usual form of the equation
of motion of an electron in an external electromagnetic field, with m being the
rest-mass of the electron and fa F ba t (Fa et + Fbadv) being the external
field.