(Solved): please show your work Review of a capacitor partially filled with dielectric material as shown in t ...

Review of a capacitor partially filled with dielectric material as shown in the drawing below. The material on the right side has a dielectric constant of $?$. a. The charge area densities $?L$ and $?R$ on the left and right side are not the same (true/false). b. The potential difference between the two plates on the left and right sides are the same, despite their different charge area densities and electric displacements. This allows us to establish a relationship between $?L$ and $?\mathrm{R}$. What is the relationship? c. The above two ideas form the foundation to work out the capacitance of this peculiar capacitor. What is the capacitance. Go through the mathematics process, especially those skipped in our class, to derive the formula. d. There will be a force on dielectric material. The best way to calculate this force is based on energy consideration. Energy stored in the capacitor is $E=\frac{Q^2}{2C}$. Force can be calculated from $F=?\frac{dE}{dx}=$ $?\frac{dE}{dC}\frac{dC}{dx}$. Prove that the force is in the form of $?\frac{{?}_0?e_{e}W}{2d}{V}^2$.