Question 35: AC fundamentals

Question

The capacitance of a parallel plate capacitor is $$400 \mathrm{~pF}$$. It is connected to an ac source of $$100 \mathrm{~V}$$ having an angular frequency $$100 \mathrm{~rad~s}^{-1}$$. If the rms value of the current is $$4 \mu \mathrm{A}$$, the displacement current is:

Accepted Answer

To determine the displacement current for a parallel plate capacitor connected to an AC source, we utilize the relationship between the capacitor's current, the voltage, and the given angular frequency.
The capacitive reactance, $$X_C$$, of the capacitor can be calculated using the formula:
$$ X_C = \frac{1}{\omega C} $$
where:
$$\omega$$ (omega) = angular frequency = $$100 \mathrm{~rad~s}^{-1}$$
$$C$$ = capacitance = $$400 \mathrm{~pF} = 400 	imes 10^{-12} \mathrm{~F}$$
Substituting the given values:
$$ X_C = \frac{1}{100 \mathrm{~rad~s}^{-1} 	imes 400 	imes 10^{-12} \mathrm{~F}} $$
$$ X_C = \frac{1}{(100 	imes 400) 	imes 10^{-12}} $$
$$ X_C = \frac{1}{40000 	imes 10^{-12}} $$
$$ X_C = \frac{1}{4 	imes 10^{-8}} $$
$$ X_C = 2.5 	imes 10^{7} \, \Omega $$
Next, we need to find the RMS current. For a capacitor in an AC circuit, the RMS voltage, $$V_{	ext{rms}}$$, and the RMS current, $$I_{	ext{rms}}$$, are related by:
$$ I_{	ext{rms}} = \frac{V_{	ext{rms}}}{X_C} $$
Given:
$$V_{	ext{rms}} = 100 \mathrm{~V}$$
Substituting the values:
$$ I_{	ext{rms}} = \frac{100 \mathrm{~V}}{2.5 	imes 10^{7} \Omega} $$
$$ I_{	ext{rms}} = 4 \mu \mathrm{A} $$
This matches the given value of the RMS current. The displacement current in an AC circuit is equivalent to the RMS current of the capacitor. Therefore, the displacement current is:
$$ 	ext{Displacement current} = 4 \mu \mathrm{A} $$
The correct answer is therefore:
Option C: 4 $$\mu$$A

Answer

$$4 	imes 10^{-2} \mu \mathrm{A}$$

Answer

0.4 $$\mu$$A

Answer

4 $$\mu$$A

Answer

4 A

AC fundamentals

The capacitance of a parallel plate capacitor is $400 \mathrm{~pF}. It is connected to an ac source of 100 \mathrm{~V} having an angular frequency 100 \mathrm{~rad~s}^{-1}. If the rms value of the current is 4 \mu \mathrm{A}$, the displacement current is:

Choose the correct answer: