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The quasi-stationary approximation is valid in regions close to the sources, i.e., at distances smaller than the typical wavelength of an electromagnetic field. In this "near-field zone" you can neglect retardation. That's why the quasitationary approximation works for usual household AC.
Again, the charge and current density of a single charge is given by
[tex]\rho(t,\vec{x})=q \delta^{(3)}[\vec{x}-\vec{y}(t)], \quad \vec{j}(t,\vec{x})=q \dot{\vec{y}}(t) \delta^{(3)}[\vec{x}-\vec{y}(t)],[/tex]
where [itex]\vec{y}(t)[/itex] is the trajectory of the particle in a fixed reference frame. You clearly see that even for a uniformly moving particle these are not stationary, and you have to use the retarded expressions to get the correct field. In that case you can also use the trick with the Lorentz boost, I've demonstrated some postings before.
Again, the charge and current density of a single charge is given by
[tex]\rho(t,\vec{x})=q \delta^{(3)}[\vec{x}-\vec{y}(t)], \quad \vec{j}(t,\vec{x})=q \dot{\vec{y}}(t) \delta^{(3)}[\vec{x}-\vec{y}(t)],[/tex]
where [itex]\vec{y}(t)[/itex] is the trajectory of the particle in a fixed reference frame. You clearly see that even for a uniformly moving particle these are not stationary, and you have to use the retarded expressions to get the correct field. In that case you can also use the trick with the Lorentz boost, I've demonstrated some postings before.