Parallel Plate Capacitor Design and Dielectric Properties Analysis

A parallel plate capacitor is designed with a dielectric material of relative permittivity $({}_r)$ equal to $6\mathrm{.}0.$ The following conditions are provided. Based on these, answer the questions below.

Given:

Plate area: $A=5\mathrm{.}0{10}^{-4}{m}^2$

Distance between plates: $d=0\mathrm{.}002m$

Applied voltage: V$=10$ V

Permittivity of free space: ${}_0=8\mathrm{.}85{10}^{-12}\frac{F}{m}$

Questions:

a$.$ Calculate the capacitance $C$ of the capacitor.

b$.$ Determine the charge $Q$ stored on each plate.

c$.$ Compute the dielectric displacement $D.$

d$.$ Calculate the polarization $P$ of the dielectric material.

e$.$ If the dielectric strength $E_{bd}=5{10}^6\frac{V}{m},$ calculate the maximum voltage the capacitor can withstand without breakdown.

f$.$ if the dielectric material that has a relative permittivity $({}_r)$ dependent on frequency as below.

Frequency$-$dependent relative permittivity $({}_r)$ :

$1Hz$:${}_r=6\mathrm{.}0$

${10}^{6}Hz$:${}_r=4\mathrm{.}5$

${10}^{12}Hz$:${}_r=3\mathrm{.}8$

${10}^{16}Hz$:${}_r=3\mathrm{.}5$

Calculate C and Q at each frequency.

g$.$ Explain how the frequency of the applied field affects each polarization mechanism and discuss why certain mechanisms dominate at specific frequency ranges

h $.$ if this material shows temperature$-$dependent relative permittivity ${}_r=6\mathrm{.}0($at $25C),{}_r=4\mathrm{.}0($at $125C),$ Curie temperature $(T_C)$:$130C,$ Describe what happens to the material's dielectric properties as it transitions through the Curie temperature.

i$.$ Draw the polarization hysteresis loops at $25C$ and $175C.$ Assume the dielectric constant has a linear relationship with temperature. Note: At $175C($above the Curie temperature $T_C=130C),$ the material is in the paraelectric phase and exhibits a linear $P-E$ relationship.