Ac Circuits

Step 1: Understanding the Concept:
In an alternating current (AC) circuit containing only a capacitor, the flow of current is related to the rate of change of voltage across the capacitor. This relationship causes a phase difference between the current and voltage waveforms.
Step 2: Detailed Explanation:
Let the alternating voltage applied to the circuit be .
The charge on the capacitor at any instant is .
The current in the circuit is the rate of flow of charge, so we differentiate the charge with respect to time:
To compare the phase of current and voltage, we express the current in terms of a sine function:
Since , we can write:
Where is the peak current.
Comparing the phase of voltage, , with the phase of current, , we find the phase difference:
A phase difference of radians is equal to 90°. The positive sign indicates that the current leads the voltage by 90°.
Step 3: Final Answer:
The phase-difference between current and voltage in a purely capacitive AC circuit is 90° (or radians), with the current leading the voltage. Therefore, option (B) is correct.

Step 1: Formula for resonance frequency in an L-C circuit.
The resonance frequency of an L-C circuit is given by the formula: where is the inductance and is the capacitance. This frequency occurs when the inductive reactance equals the capacitive reactance.
Step 2: Conclusion.
Thus, the correct expression for the resonance frequency is .

Step 1: Power factor in an AC circuit.
The power factor ( ) in an alternating current (AC) circuit is the cosine of the phase angle ( ) between the current and voltage. For purely inductive circuits, the phase angle between current and voltage is .
Step 2: Power factor for purely inductive circuits.
The power factor in a purely inductive circuit is given by:
Step 3: Analyze options.
- (A) 0: Correct. In a purely inductive circuit, the power factor is zero because current lags the voltage by 90° and no real power is transferred.
- (B) 1: Incorrect. A power factor of 1 happens only in a purely resistive circuit.
- (C) 0.5: Incorrect. This value does not apply to purely inductive circuits.
- (D) infinity: Incorrect. This is not a valid power factor value in any circuit.
Step 4: Conclusion.
In a purely inductive circuit, the power factor is 0.

Step 1: Understanding the phase difference in a capacitor.
In an alternating current (a.c.) circuit containing only a capacitor, the current leads the voltage by 90°. This is because the voltage across the capacitor lags the current due to the charging and discharging of the capacitor.
Step 2: Conclusion.
Thus, the phase difference between the current and voltage in a purely capacitive circuit is 90°.

In an alternating current (AC) circuit, the impedance is given by the relation: Where: - is the resistance of the circuit, - is the inductive reactance, - is the capacitive reactance. Here, we are given that: - , - . Since , their difference is zero. Therefore, the impedance simplifies to: Substituting the given value of : Now, the root mean square (RMS) value of the voltage is given by the formula: Where . So, Thus, the impedance of the circuit is , and the RMS value of the voltage is approximately .