Where is electromagnetic induction in the Jefimenko equations?

Question

I'm currently exploring the Jefimenko Equations and practicing using them to find things like the electric field from a particle or the magnetic field around a current. In general, I've read that the Jefimenko Equations are an alternative to the Maxwell Equations. However, one thing that I can't seem to figure out is electromagnetic induction:

$$\nabla\times \boldsymbol{E} = -\frac{\partial \boldsymbol{B}}{\partial t}$$

From this Maxwell Equation, we can figure out the voltage induced in a wire that has a changing magnetic field through it using Stokes' Theorem. However, the Jefimenko equation for $\boldsymbol{E}(\boldsymbol{r},t)$ does not contain a term for a changing magnetic field. So how do the equations explain the induced voltage in a wire from a changing magnetic field using Jefimenko's Equations?

Answer 1

Ignatowski's equations follow from the representation of the fields by the vector and scalars as $$\begin{align}\mathbf{E}&=-\nabla \phi - \frac{\partial \mathbf{A}}{\partial t} \tag{1}\label{1}\\ \mathbf{B}&= \nabla \times\mathbf A \tag{2}\label{2} \end{align}$$ These two represent Faraday's induction law.

When you substitute the integrals into $\eqref{1}$ and$\eqref{2}$ $$\begin{align} \mathbf A &= \frac{\mu_0}{ 4\pi}\int \frac{[\mathbf J]}{R}dV \tag{3}\label{3}\\ \phi &= \frac{1}{ 4\pi \epsilon_0 } \int\frac{[\rho]}{R}dV \tag{4}\label{4} \end{align}$$ you recover Ignatowski's equations. For details see McDonald: "The Relation Between Expressions for Time-Dependent Electromagnetic Fields Given by Jefimenko and by Panofsky and Phillips", Am. J. Phys. 65 (11), November 1997