A point charge `q` moves with a non-relatives velocity v = const. Find the displacement current density `dotd` at a point location at a distance `r` from the charges on a straight line
(a) coinciding with the charge path,
(b) perpendicualr to the path and passing through the charge.

Explanation

In the non-relativistic limit.
`vec(E) = (q)/(4pi epsilon_(0) r^(3)) vec(r)`
(a) On a striaght line coinciding with the charge path,
`vec(j_(4)) = epsilon_(0) (del vec(E))/(del t) = (q)/(4pi) [(-vec(V))/(r^(3)) - (3 vec(r) r)/(r^(4))], ("using", (d vec(r))/(dt) = -vec(v).)`
But in this case, `dotr = -v` and `v (vec(r))/(r) = vec(v)`, so, `j_(d) = (2q vec(v))/(4pi r^(3))`
(b) In this case, `dotr = 0`, as, `vec(r) _|_ vec(v)`. Thus,
`j_(4) = (q vec(v))/(4pi r^(3))`