Solution
= Solution
The temporal component found in part (a) gives directly
$$
\boxed{
\mathcal E(t)-\mathcal E(0)
=q\int_0^t\mathbf E\cdot\mathbf v(s)\,ds
}.
$$
For a constant <electric field>, this is the work done by the field along the particle trajectory:
$$
\boxed{
\mathcal E(t)=\mathcal E(0)
+q\mathbf E\cdot[\mathbf x(t)-\mathbf x(0)]
}.
$$