Which of the following is true for an ideal capacitor connected to a sinusoidal voltage source ?

Options

A. neither the avarage power nor the avarage nor the avarage current is zero
B. avarage voltage is zero but the avarage power is non zero
C. both avarage power and avarage current is zero
D. Avarage power is zero but the avarage current is non zero

Answer

To solve the question regarding the behavior of an ideal capacitor connected to a sinusoidal voltage source, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Circuit**: We have an ideal capacitor \( C \) connected to a sinusoidal voltage source. The voltage can be expressed as:
\[ V(t) = V_0 \sin(\omega t + \phi) \]
where \( V_0 \) is the peak voltage, \( \omega \) is the angular frequency, and \( \phi \) is the phase angle.

2. **Capacitive Reactance**: The capacitive reactance \( X_C \) is given by:
\[ X_C = \frac{1}{\omega C} \]
This represents the opposition that the capacitor offers to the alternating current.

3. **Impedance of the Capacitor**: The impedance \( Z \) of the capacitor is purely imaginary and can be expressed as:
\[ Z = -jX_C \]

4. **Calculate the Current**: The current \( I \) through the capacitor can be calculated using Ohm's law:
\[ I = \frac{V}{Z} \]
The magnitude of the current can be expressed as:
\[ |I| = \frac{V_0}{|Z|} = \frac{V_0}{X_C} \]

5. **Phase Relationship**: The current leads the voltage by \( 90^\circ \) (or \( \frac{\pi}{2} \) radians). This can be derived from the fact that the phase angle \( \phi \) for a capacitor is:
\[ \tan(\phi) = \frac{-jX_C}{0} \quad \Rightarrow \quad \phi = -\frac{\pi}{2} \]
Thus, the current can be expressed as:
\[ I(t) = I_0 \sin(\omega t + \frac{\pi}{2}) = I_0 \cos(\omega t) \]

6. **Average Power Calculation**: The average power \( P_{\text{avg}} \) in an AC circuit is given by:
\[ P_{\text{avg}} = V_{\text{rms}} \cdot I_{\text{rms}} \cdot \cos(\phi) \]
Since \( \phi = -\frac{\pi}{2} \), we have:
\[ \cos(-\frac{\pi}{2}) = 0 \]
Therefore, the average power is:
\[ P_{\text{avg}} = 0 \]

7. **Average Current and Voltage**: The average current and voltage over a complete cycle for a sinusoidal waveform is also zero:
\[ I_{\text{avg}} = 0 \quad \text{and} \quad V_{\text{avg}} = 0 \]

### Conclusion:
Based on the calculations, we find that both the average power and the average current are zero for an ideal capacitor connected to a sinusoidal voltage source. Therefore, the correct statement is:

**The average power and average current are both zero.**