Let us consider these two Maxwell equations:
$$\frac{\partial \vec{B}}{\partial t}=-\vec{\nabla}\times \vec{E}$$ and
$$\frac{\partial \vec{E}}{\partial t}=\frac{1}{\epsilon_0}\left(-\vec{J}+\frac{1}{\mu_0}\vec{\nabla}\times \vec{B}\right).$$
When we consider faraday's law of induction, we usually assume that the changes are slow, and thus we can neglect radiation by assuming that the left hand side of the second equation is zero. That is, a changing current creates a changing magnetic field, which in turn creates a changing electric field, per the first equation.
I cannot understand this. First, if we can neglect the change in E from the second equation, should not we also neglect the change in B in the first equation? Second, this imply that we can have changing electric and magnetic fields that are not electromagnetic waves. But are not all changing magnetic or electric fields EM waves? or is this approximation equivalent to a charge moving at constant speed, in which the change in E and B are not due to radiation but just to the translation motion of the static field lines
