Two parallel metal plates, each of area 200 cm², stand 2.0 mm apart…

Two parallel metal plates, each of area $200\ \text{cm}^2$, stand $2.0\ \text{mm}$ apart. The gap is filled with a layered plastic whose dielectric constant grows steadily across the gap:

where $x$ is the distance from plate $P$ and $d$ is the gap, as in the figure. So $K$ is $2.0$ next to $P$ and $4.0$ next to plate $Q$. The plastic touches both plates, with no air left between. Free charge is the charge on the metal; bound charge is the charge that shows on the plastic when its molecules are polarised. A $100\ \text{V}$ battery holds $P$ positive and $Q$ negative. Ignore the effects at the edges of the plates. Take $\varepsilon_0 = 8.85 \times 10^{-12}\ \text{C}^2\,\text{N}^{-1}\,\text{m}^{-2}$.

1. What is the capacitance?
2. What is the free charge on plate $P$?
3. What is the electric field in the plastic just next to plate $P$?
4. What bound charge sits on the face of the plastic that touches $P$, with its sign?
5. What bound charge sits on the face that touches $Q$, with its sign?
6. The plastic also holds bound charge spread through its volume. What is its total, with its sign?