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

Question

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)

The Maxwell-Faraday Equation reads:

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

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?

Answer 1

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.