I have a capacitor with round plates of radius $r_{2}$ kept at a distance $d$, with a voltage $V\left(t\right) = V_{0} \cos \left(\omega t\right).$ Inside, there is cylindrical material with a certain dielectric constant ($\epsilon_{r}$) and magnetic permittivity ($\mu_{r}$), which occupies all the distance between the plates, but has a radius $r_{1} \lt r_{2}.$
How do you evaluate the displacement current?
The solution states it is enough to compute the derivative of the electric field and multiply it by the dielectric constant in vacuum ($\epsilon_{0}$), but I thought the definition of $\mathbf{J}_{displacement}$ was the time derivative of the vector $\mathbf{D}$, which has a different value in different areas of the capacitor; in fact, the electric field must be constant inside the capacitor so that the voltage between the plates is univocal in all points of the plates, and the vector $\mathbf{D}$ thus has a value $\mathbf{D} = \epsilon_{0} \mathbf{E}\left(t\right)$ outside the material and $\epsilon_{0}\epsilon_{r} \mathbf{E} \left(t\right)$ inside of it. How can the displacement current be the same in all the space?
Furthermore, I also have a doubt about the value of the magnetic field inside the capacitor.
I would apply Ampere's law in matter, which states that the circuitation of $\mathbf{H}=\frac{\mathbf{B}}{\left(\mu_{r}\mu_{0}\right)}$ is equal to the current inside the loop, $i.$ I would than find the value of $i$ using the above information about $\mathbf{J}_{displacement}$ and consider $\mu_{r}$ when studying the field inside the material. It looks like it shouldn't be done like this: $\mu_{r}$ is never taken into consideration. Why?
