Question 47: Electrostats & Capacitance

Question

A parallel plate capacitor having a dielectric constant 5 and dielectric strength $$10^6 \mathrm{~V} \mathrm{~m}^{-1}$$ is to be designed with voltage rating of $$2 \mathrm{~kV}$$. The field should never exceed $$10 \%$$ of its dielectric strength. To have the capacitance of $$60 \mathrm{~pF}$$ the minimum area of the plates should be

Accepted Answer

To design the parallel plate capacitor with the given specifications, we need to determine the minimum area of the plates. LetÃ¢ÂÂs break down the given information and use it step by step.
The capacitor has a dielectric constant $$\kappa = 5$$ and a dielectric strength of $$10^6 \mathrm{~V} \mathrm{~m}^{-1}$$. The voltage rating is $$2 \mathrm{~kV}$$ and the maximum electric field allowed is $$10 \%$$ of its dielectric strength.
First, calculate the maximum electric field:
$$E_{	ext{max}} = 0.1 	imes 10^6 \mathrm{~V} \mathrm{~m}^{-1} = 10^5 \mathrm{~V} \mathrm{~m}^{-1}$$
Given the voltage rating, the separation between the plates (d) can be found using the relation:
$$V = E_{	ext{max}} 	imes d$$
Rearranging for $d$:
$$d = \frac{V}{E_{	ext{max}}} = \frac{2000 \mathrm{~V}}{10^5 \mathrm{~V} \mathrm{~m}^{-1}} = 0.02 \mathrm{~m}$$
Next, we use the formula for the capacitance of a parallel plate capacitor:
$$C = \kappa \epsilon_0 \frac{A}{d}$$
Where:
$$C = 60 \mathrm{~pF} = 60 	imes 10^{-12} \mathrm{~F}$$
$$\kappa = 5$$
$$\epsilon_0 = 8.85 	imes 10^{-12} \mathrm{~F m}^{-1}$$
$$d = 0.02 \mathrm{~m}$$
$A = 	ext{Area of the plates}$
Rearranging for $A$:
$$A = \frac{C 	imes d}{\kappa 	imes \epsilon_0}$$
Substituting the values:
$$A = \frac{60 	imes 10^{-12} \mathrm{~F} 	imes 0.02 \mathrm{~m}}{5 	imes 8.85 	imes 10^{-12} \mathrm{~F m}^{-1}} = \frac{60 	imes 0.02}{5 	imes 8.85} 	imes 10^{-12 + 12} \mathrm{~m}^2 = \frac{1.2}{44.25} \mathrm{~m}^2$$
Simplifying further, we get:
$$A \approx 2.71 	imes 10^{-2} \mathrm{~m}^2$$
Therefore, the minimum area of the plates should be:
Option B: $$2.7 	imes 10^{-2} \mathrm{~m}^2$$

Answer

$$27.1 	imes 10^{-4} \mathrm{~m}^2$$

Answer

$$2.7 	imes 10^{-2} \mathrm{~m}^2$$

Answer

$$2.71 	imes 10^{-4} \mathrm{~m}^2$$

Answer

$$27.1 	imes 10^{-2} \mathrm{~m}^2$$

Electrostats & Capacitance

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