# Where is electromagnetic induction in the Jefimenko equations? 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?

• As a note, I suspect that L. V. Lorenz published "Jefimenko" equations already in about 1860. Aug 25, 2020 at 22:58
• If the magnetic field is changing, a current is changing to produce it, which is why the electric field has a $\partial \mathbf{J}/\partial t$ term. Aug 25, 2020 at 22:59
• @Javier, I believe that's the term for electromagnetic waves right? I thought induction was different than EM radiation? Also, how would it explain the induced voltage from a moving magnet?
– nreh
Aug 25, 2020 at 23:08

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
• for any solenoidal vector field $\mathbf{v}$, that is if $\nabla\cdot \mathbf{v}=0$ can be written as $\mathbf{v} = \nabla \times \mathbf{a}$ for some $\mathbf{a}$; therefore $\nabla \times \mathbf{E}+\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{E}+\frac{\partial \mathbf{A}}{\partial t}=0.$ That is the vector field $\mathbf{E}+\frac {\partial \mathbf{A} }{\partial t}$ is lamellar and can be written as the gradient of a scalar field: $\mathbf{E}+\frac {\partial \mathbf{A}} {\partial t} = -\nabla \phi$ .... Aug 26, 2020 at 16:49
• The scalar $\phi$ and vector potentials $\mathbf{A}$ are directly related to charge $\rho$ and current $\mathbf{J}$ densities, resp., see (3) and (4), and Ignatowski's equations are that relationship after differentiating the integrals with appropriate precautions taken for retardation in time. Aug 26, 2020 at 16:57