How did this field magically appear?
Induced electric field is caused by the variation of current-density in the stationary loop and not by magic. The electric and magnetic fields are correlated by
$$\mathbf \nabla \times \mathbf E~=~ -\partial_t\mathbf B\;.$$
I am also aware that a current carrying wire exerts a magnetic force on a moving charge due to length contraction, responsible for creating an extra electrostatic force on the charge. Can relativity be used to explain Faraday's law as well?
I could conclude that you were saying A magnetic field is caused by length contraction and....
Well,
Magnetic field is due to the relativistic effect of electric field.
But in the same way,
Electric field is due to the relativistic effect of magnetic field.
That is, taking one field as the mean the charges interact, the relativistic transformation equations make the presence of the other field imperative.
But, both electric and magnetic field can't be relativistic effect at once.
As Jefimenko in his paper points out:
The only correct interpretation must be that interactions between electric charges that are either entirely velocity independent or entirely velocity dependent is incompatible with the relativity theory.Both fields—the electric field (producing a force independent of the velocity of the charge experiencing the force) and the magnetic field (producing a force
dependent on the velocity of the charge experiencing
the force)—are necessary to make interactions between
electric charges relativistically correct. By inference
then, any force field compatible with the relativity theory
must have an electric-like ‘subfield’ and a magnetic-like
‘subfield’.
The electric and magnetic fields are the two aspects of the electromagnetic field. It depends on frame whether electromagnetic field would like an electric field or magnetic field or combination of both.
See Christoph's answer here.
Also, see the last part of Timaeus' answer here.
You are right that a changing magnetic field creates (induces) an electric field, this is an actual law of nature.
Now if you put a conductor where the magnetic field is changing, you will get a current due to the produced electric field.
But in the case of the moving conductor moving through a magnetic field the reason is different. The reason for the produced current is Lorentz force, the electrons in the conductor are pushed by Lorentz force and hence you get the current.
Notice in this case, even if the conductor is moving through the magnetic field the magnetic field is NOT changing so electric field is not produced, the reason must therefore be the Lorentz force.
Whenever you get confused just check whether the magnetic field is changing or not. If the magnetic field is changing then the reason for the current must be an induced electric field, if it is not changing the reason must be Lorentz force.
Best Answer
Yes varying electric field induce a magnetic field as stated by the Maxwell's fourth equation (an extension of the Ampère's law). In free space that is: $$ \nabla\times\vec{B}=\mu_0\vec{J}+\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t} $$ An example can be any type of electromagnetic wave, where $\vec{E}$ (and also $\vec{B}$) is varying with time, a simple one would be a plane-wave where $\vec{E}=\vec{E}_0\cos(\omega t-kz)$ with $\vec{E}_0$ is a costant vector (more here).
In a dielectric material the Maxwll's equation becomes: $$ \nabla\times\vec{H}=\vec{J}_f+\frac{\partial\vec{D}}{\partial t} $$ where $\vec{D}=\epsilon_0\vec{E}+\vec{P}$ and so $\frac{\partial\vec{D}}{\partial t}=\epsilon_0\frac{\partial\vec{E}}{\partial t}+\frac{\partial\vec{P}}{\partial t}$. The additional term $\frac{\partial\vec{P}}{\partial t}$ keeps track of any moving charge due to the polarization of the material.