Q$1)$ A simple linear actuator is made by stacking alternating conducting disks and piezoelectric disks as shown in the figure below. There are N piezoelectric disis, ench of thickness t and radius a$,$ and $N+1$ conducting plates. The conducting $$isk$\text{'}$s whole purpose is to apply an external voltage to enerau in electric field intensity in the piezoelectric disks. Given the properties of the piezoelectric material $($relative permittivity $lo{n}_{ii}$ and piezocieutric constant $d_i)$:

a$.$ Derive the displacement of the stack for an appiied voltage V $($neglect the normal component$).$

b$.$ Derive the force the stack can generate for an applied voltage V $.$

c$.$ By using the table below, calculate the displacement and force for a $33$ cut barium titanate oxide $($BaTiO$3)$ piezoeicutric with dimensions $a=10mm,t=1mm,$ and $V=36V$ for a stack with $N=40$ disks.

d$.$ What is the maximum possible displacement and force if the breakdown electric field in the crystal is $32,000\frac{V}{m}m$ whereas the breakdown voltage in air is $3000\frac{V}{m}m?$