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....
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
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.