# Faraday's Law and Lenz's Law: Is there any theoretical explanations on why changing magnetic field induces an electric field?

This is a more specific extension to this question I came across today

One certain aspect of Faraday's Law always stumped me (other than it is an experimental observation back in the 19th century)

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

I am also briefly aware that in special relativity, magnetic fields in one frame are basically electric fields in another, but

Q1 How exactly does a changing magnetic field induces an electric field. Is there any theoretical explanation that came up in the literature using more fundamental theories such as QED and relativity that explains how it happens?

Q2 Is their a theoretical reason Why does the electric field is produced in a way that opposes the change in the magnetic field?

Indeed, this observation remains mysterious from a 19th century viewpoint.

Since we know special relativity, though, it is natural in the covariant formulation of electromagnetism that spatial and temporal changes of fields are interrelated. More specifically, we need to express the three-vectors $\vec E$ and $\vec B$ in a covariant way, which is done by defining the field strength tensor $F$ component-wise as

$$F^{i0} := E^i \quad \text{and} \quad F^{ij} = \epsilon^{ijk}B_k$$

This object now behaves properly (as a 2-tensor) under Lorentz transformations, in contrast to the three-vectors $\vec E$ and $\vec B$ whose components mix.

Now, Maxwell's equations of course must also be written covariantly,

$$\partial_\mu F^{\mu\nu} = j^\nu \quad \text{and} \quad \partial_\mu\left(\frac{1}{2}\epsilon^{\mu\nu\sigma\rho}F_{\sigma\rho}\right) = 0$$

and if you go and write out this with $\partial_t$ and $\vec \nabla$ and so on again, you get back, among others, the Maxwells-Faraday equation.

So, essentially, the mixture of electric and magnetic fields, and their spatial and temporal changes, is a direct consequence of the fact that the world is not Galilean, but relativistic.

• So you mean because the world is relativistic, electromagnetism is from an object known as the field strength tensor. And when this tensor varies in space and time (which is what happens when we take $\nabla$ and $\partial_t$ in the above equation), it manifest as two objects known as the electric and magnetic fields, which is why changing magnetic fields can induce electric fields because they are just part of the field strength tensor? Mar 6 '15 at 11:05
• @Secret: Yes, exactly! Mar 6 '15 at 11:08
• That doesn't seem to answer the question. All you've said, in efffect, is, "If you write it in terms of relativity you get the same relationship." But what's the physical mechanism?
– pwf
Dec 19 '16 at 5:49
• @pwf I believe ACuriousMind has said more than that. SR has other justifications aside from and quite independently of EM, even though EM features centrally in Einstein's studies of SR. So what is being said here is, given SR's other justifications, let's see what it has to say about the allowable structure of Maxwell's equations. Moreover, when you ask for a "physical" mechanism, well, when you're dealing with E and B separately, without recognizing their unavoidable unification, you're physical questions about only part of the physical object. There's no mechanism: one does not give rise ... Dec 19 '16 at 10:18
• @pwf ... the other as the separated Faraday and Ampère laws tend to make us think: they both just are, different parts of the underlying physical object. Dec 19 '16 at 10:20