Why can't a current carrying loop (curl of the electric field exists) produce a time varying magnetic field? If a time varying magnetic field can give value to the curl of an electric field, why not the other way round?
That is, why can't an enclosed loop with some emf produced (basically a current carrying loop) produce a changing magnetic field?
It does produce a constant magnetic field, yes. But according to Faraday's law, curl E=-dB/dt
if the curl of the electric field has a value, shouldn't the time derivative also have a value? Meaning: a changing magnetic field should be produced.
 A: In a wire carrying a steady current, there is no electric field if the resistance of the wire is zero (i.e. a superconductor). If you make a loop of such wire and induce a current in it, then there is no electric field and the closed line integral of the electric field is zero.
In a wire with finite resistance, there is an electric field and on the face of it, there would be a finite line integral going around a closed loop. However, somewhere in that circuit, there must be an EMF source that has an exactly equal and opposite line integral (if the current is steady).
The net result is a closed line integral of zero (the circuital law) and no changing magnetic field due to steady current.
A: I may be missing your point, but current carrying loops produce varying magnetic fields in all kinds of situations:  AC motors, transformers, inductors, demagnifiers, and many others.  In an electromagnetic wave, the interaction of varying electric and magnetic fields determines the rate of propagation of the wave.
