Why a transformer cannot use DC (Ampere's law)?

[kekkeroo]

A direct current in the primary winding results in a constant magnetic field going through the center of the coil. If we have a magnetic core and a second winding, why doesn't the magnetic flux going through the core result in a current in the secondary coil, as well, according to Ampere's law?

 

[Baluncore]

Welcome to PF.

The short answer is that the changing current in the primary causes a changing flux in the core.
The voltage in the secondary is proportional to the rate of change of flux in the core.
No change in core flux, implies no secondary voltage, so no secondary circuit current.
https://en.wikipedia.org/wiki/Transformer

 

[Rive]

why doesn't the magnetic flux going through the core result in a current in the secondary coil

Why do you need a generator spinning to have current in its coil?