Question 48: Electrostatics & Capacitance

Question

Figure below shows a network of resistors, cells, and a capacitor at steady state.
What is the current through the resistance 4 $$\Omega$$ ?

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

1.0 A

Answer

0.2 A

Answer

Zero

Answer

0.5 A

Electrostatics & Capacitance

Figure below shows a network of resistors, cells, and a capacitor at steady state. What is the current through the resistance 4 $\Omega$ ?

Choose the correct answer: