In Feynman's Lectures on Physics, volume 2, Chapter 18 (18-5), we look at the creation of a electromagnetic field, due to a moving infinite sheet of charge,
By looking at the Maxwell equations alone, it is not easy to see directly how to get the solution. So we will first show you what the answer is and then verify that it does indeed satisfy the equations. The answer is the following: The field B that we computed is, in fact, generated right next to the current sheet (for small x). It must be so, because if we make a tiny loop around the sheet, there is no room for any electric flux to go through it $^*$. But the field B out farther—for larger x—is, at first, zero. It stays zero for awhile, and then suddenly turns on. In short, we turn on the current and the magnetic field immediately next to it turns on to a constant value B ; then the turning on of B spreads out from the source region. After a certain time, there is a uniform magnetic field everywhere out to some value x, and then zero beyond. Because of the symmetry, it spreads in both the plus and minus x-directions.
$^*$ What does he mean, by "there's no room for electric flux to go through" , why does it conclude, "magnetic field immediately next to it turns on to a constant value B"? How can this be, how can magnetic field suddenly attain a constant value, it has to increase from 0 shouldn't it ?