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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?

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  • $\begingroup$ As a note, I suspect that L. V. Lorenz published "Jefimenko" equations already in about 1860. $\endgroup$
    – my2cts
    Aug 25, 2020 at 22:58
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    $\begingroup$ 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. $\endgroup$
    – Javier
    Aug 25, 2020 at 22:59
  • $\begingroup$ @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? $\endgroup$
    – nreh
    Aug 25, 2020 at 23:08

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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

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  • $\begingroup$ So I'm kind of a noob when it comes to electromagnetism and the mathematics behind it. As a result, I'm struggling to see how the derivation of the equations would give us a method of calculating the induced voltage in a wire using only the two equations. Could you explain to me where the term for induction would be in the Ignatowski's/Jefimenko's equations (if there is one)? Am I just misunderstanding what Ignatowski's equations are? $\endgroup$
    – nreh
    Aug 25, 2020 at 23:46
  • $\begingroup$ 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: $\mathbf{E}+\frac {\partial \mathbf{A}} {\partial t} = -\nabla \phi$ .... $\endgroup$
    – hyportnex
    Aug 26, 2020 at 16:49
  • $\begingroup$ 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. $\endgroup$
    – hyportnex
    Aug 26, 2020 at 16:57

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