Question

A voltage \( v = v_0 \sin \omega t \) applied to a circuit drives a current \( i = i_0 \sin (\omega t + \phi) \) in the circuit. The average power consumed in the circuit over a cycle is:

• A. Zero
• B. \( i_0 v_0 \cos \phi \)
• C. \( \frac{i_0 v_0}{2} \)
• D. \( \frac{i_0 v_0}{2} \cos \phi \)

Hint:

In AC circuits, the power factor \( \cos \phi \) determines the actual power consumption. It accounts for phase differences between voltage and current.

Answer 1

The Correct Option is D

The average power consumed in an AC circuit is given by: \[ P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos \phi \] Since the peak voltage and current are related to their rms values as: \[ V_{\text{rms}} = \frac{v_0}{\sqrt{2}}, \quad I_{\text{rms}} = \frac{i_0}{\sqrt{2}} \] Substituting these into the power formula: \[ P_{\text{avg}} = \left(\frac{v_0}{\sqrt{2}}\right) \left(\frac{i_0}{\sqrt{2}}\right) \cos \phi \] \[ P_{\text{avg}} = \frac{i_0 v_0}{2} \cos \phi \] Thus, the correct answer is (2).