Question

A parallel plate capacitor has a separation between plates of 0.885 mm. It has a capacitance of $1\,\mu\text{F}$ when the space between the plates is filled with an insulating material of resistivity $1 \times 10^{13}\,\Omega\,\text{m}$ and resistance $17.7 \times 10^{14}\,\Omega$. The relative permittivity of the insulating material is $\alpha \times 10^7$. The value of $\alpha$ is \underline{\hspace{2cm}}.

Answer 1

Correct Answer: 2

Step 1: Understanding the Concept:
The insulating material acts both as a dielectric for the capacitor and as a resistor to leakage current. We use the resistance properties to find the physical dimensions (Area $A$) of the plates. Once the area is known, we use the capacitance formula to find the relative permittivity $\epsilon_r$.
Step 2: Key Formula or Approach:
1. Resistance: $R = \rho \frac{d}{A} \implies A = \frac{\rho d}{R}$.
2. Capacitance: $C = \frac{\epsilon_r \epsilon_0 A}{d} \implies \epsilon_r = \frac{C d}{\epsilon_0 A}$.
Step 3: Detailed Explanation:
Given values:
Separation $d = 0.885\text{ mm} = 0.885 \times 10^{-3}\text{ m}$.
Capacitance $C = 1 \mu\text{F} = 10^{-6}\text{ F}$.
Resistivity $\rho = 1 \times 10^{13} \Omega\text{ m}$.
Resistance $R = 17.7 \times 10^{14} \Omega$.
First, calculate the cross-sectional area $A$ using the resistance formula:
$A = \frac{\rho \times d}{R} = \frac{(1 \times 10^{13}) \times (0.885 \times 10^{-3})}{17.7 \times 10^{14}}$.
$A = \frac{0.885 \times 10^{10}}{17.7 \times 10^{14}} = \frac{0.885}{17.7} \times 10^{-4}$.
Since $17.7 = 2 \times 8.85 = 20 \times 0.885$:
$A = \frac{1}{20} \times 10^{-4} = 0.05 \times 10^{-4} = 5 \times 10^{-6}\text{ m}^2$.
Now, calculate the relative permittivity $\epsilon_r$ using the capacitance formula:
$C = \frac{\epsilon_r \epsilon_0 A}{d} \implies \epsilon_r = \frac{C \times d}{\epsilon_0 A}$.
$\epsilon_r = \frac{10^{-6} \times 0.885 \times 10^{-3}}{(8.85 \times 10^{-12}) \times (5 \times 10^{-6})}$.
$\epsilon_r = \frac{0.885 \times 10^{-9}}{44.25 \times 10^{-18}}$.
$\epsilon_r = \frac{0.885}{44.25} \times 10^9$.
Notice that $44.25 = 5 \times 8.85 = 50 \times 0.885$:
$\epsilon_r = \frac{1}{50} \times 10^9 = 0.02 \times 10^9 = 2 \times 10^7$.
The problem states $\epsilon_r = \alpha \times 10^7$.
Comparing the expressions, $\alpha = 2$.
Step 4: Final Answer:
The value of $\alpha$ is 2.