What are electromagnetic potentials of moving particle?

Question

So, we've got the vector of position of a point charge $q$

$\vec{r}_0(t)=(A\sin(\omega t),0,0)$

We can easily determine the charge and current density

$\rho=q\,\delta^3(\vec{r}-\vec{r}_0(t))$

$\vec{j}=(qA\omega\cos(\omega t)\,\delta^3(\vec{r}-\vec{r}_0(t)),0,0)$

Using Maxwell equation in potentials for Lorentz gauge

$\square\varphi = \frac{\rho}{\varepsilon_0}$

If all of my assumptions are right

$\varphi=\frac{q}{4\pi\varepsilon_0|\vec{r}-\vec{r}_0(t)|}$

Same for magnetic potential for x coordinate (for y and z it's obviously $A_y=A_z=0$)

$\square A_x = \mu_0 qA\omega\cos(\omega t)\,\delta^3(\vec{r}-\vec{r}_0(t))$

Can I do by the same logic?

$A_x=\frac{\mu_0 qA\omega\cos(\omega t)}{4\pi|\vec{r}-\vec{r}_0(t)|}$

If not, how do I define the field?

Answer 1

Replace $t$ with $t^{ret} = t - \frac{|\mathbf{r} -\mathbf{r}_0|}{c}$ in your solutions, and you're right.

You can prove it using Green's function for wave equation in free 3D space.

$\varphi (\mathbf{r},t)=\frac{q }{4\pi \varepsilon_0 \left|\vec{r}-\vec{r}_0\left(t - \frac{|\mathbf{r} -\mathbf{r}_0|}{c}\right)\right|}$$\qquad, \qquad$ $A_x (\mathbf{r},t) = \frac{\mu_0 qA\omega\cos \left(\omega \left( t - \frac{|\mathbf{r} -\mathbf{r}_0|}{c}\right)\right)}{4\pi\left|\vec{r}-\vec{r}_0\left(t - \frac{|\mathbf{r} -\mathbf{r}_0|}{c}\right)\right|}$

You can find here https://en.wikipedia.org/wiki/Green%27s_function a list of expressions of Green's function for several equations.