 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 $I=dQ/dt$, soThe 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: $[\omega p_0/d]={\mathrm{s}}^{-1}\mathrm{C}\mathrm{m}/\mathrm{m}=\mathrm{A}$.