When an electrostatic field is created, doesn't it automatically generate an induced magnetic field?

Question

I know that an electrostatic (or magnetostatic) field does not depend on time but I was imagining to put a charge in a region of space. First there was no electrostatic field and then it appears, how does it all happen? There is a variation of the electrostatic field in the instant in which I create the aforementioned field, from 0 to a specific value. This variation is time dependant, so shouldn't it induce a magnetic field according to Ampere-Maxwell's law? I feel like I am missing something.

Answer 1

I was imagining to put a charge in a region of space. First there was no electrostatic field and then it appears, how does it all happen?

This is not possible. It would violate the conservation of charge.

Instead, what can happen is that you can start with no charge and then you can make a pair of equal and opposite charges that are initially superimposed and then separate. The charges must move to separate, so it is not electrostatic. There is electromagnetic radiation, with the leading edge being standard dipole radiation

Answer 2

There is a magnetic field while the electric field is being produced, but the magnetic field goes to zero when the electric field becomes static.

Answer 3

You get an electrostatic field by separating charges, i.e. by moving electrons. You get a magnetostatic field by aligning the magnetic dipoles of the electrons, i.e. exactly the state that exists in a permanent magnet.

When an electrostatic field is created, doesn't it automatically generate an induced magnetic field?

We are talking about static fields, so we are neglecting the generation of these states. Then the answer is no. Both fields are absolutely independent of each other and are based solely on the fact that charges are equipped with the two intrinsic properties of an electric and a magnetic field.