Oscillating current
= Oscillating current
Choose $p(t)=p_0\cos\omega t$ along $+z$. If the end charges are $\pm Q(t)$ separated by fixed distance $d$, then $p(t)=Q(t)d$. The conventional current into the positive end is $I=dQ/dt$, so
$$
\boxed{I(t)=\frac{\dot p(t)}d=-\frac{\omega p_0}{d}\sin\omega t,\qquad I_{\max}=\frac{\omega p_0}{d}.}
$$
The current is in <phase quadrature> with the dipole moment. A different time origin or current orientation changes the sine's sign or phase but not its amplitude. Units check: $[\omega p_0/d]=\mathrm{s^{-1}}\mathrm{C\,m}/\mathrm m=\mathrm A$.