The problem: A cylindrical thin shell of electric charge has length $l$ and radius $a$, where $l \gg a$. The surface charge density on the shell is $\sigma$. The shell rotates about its axis with angular velocity $\omega = kt$, where $k$ is a constant and $t \geq 0$. (The problem can be found here.)
I am then asked to find the magnetic field inside the cylinder. Of course we need to find the current distribution and use Ampére's law, and I have no trouble doing that and obtaining the same result as in the above link. However, since this is not a magnetostatic situation (the surface current density $K = \sigma akt$ is time-dependent), Ampére's law is not a priori valid without displacement current. How does one argue that, in fact, the electric field is static? Of course, the charge distribution is static, and from this it follows that the time derivative of the electric field is divergenceless, but that doesn't seem to be enough.
Best Answer
The electric field inside the cylinder won't be exactly static, because the charges are accelerating in the direction perpendicular to their radius vectors and hence they will produce induced electric field whose lines of force will turn in circles inside the cylinder, the highest field being near the cylinder wall.
So the Ampere law is not really applicable, because there will be some displacement current which it neglects.
However, a good estimate of the magnetic field, even when there is some displacement current present, is usually possible to obtain by using the Biot-Savart law. For cylinder with surface current, it turns out that the magnetic field inside calculated that way is uniform and along the axis of symmetry.
Once you obtain that estimate of magnetic field value as a function of time, you can estimate the electric field inside the cylinder as well, from the Maxwell equation
$$ \oint \mathbf E\cdot d\mathbf s = -\frac{d\Phi_B}{dt}. $$
Then you can calculate flux of displacement current through any rectangle passing through the cylinder and apply the Maxwell-Ampere law to determine the correction to magnetic field at every distance from the axis.