Question

In an A.C. circuit, potential difference and current are given as, \[ V = 100 \sin(100t) \, \text{volts}, \quad i = 100 \sin\left(100t + \frac{\pi}{3}\right) \, \text{mA}. \] The power consumed in the circuit is:

• A. $10^{4}$ watt
• B. 10 watt
• C. 2.5 watt
• D. 5 watt

Hint:

In AC circuits, average power is given by $P = V_{rms} \cdot I_{rms} \cdot \cos\phi$, where $\phi$ is the phase difference.

Answer 1

The Correct Option is C

Step 1: Identify RMS values.
For voltage: \[ V_m = 100 \quad \Rightarrow \quad V_{rms} = \frac{100}{\sqrt{2}}. \] For current: \[ I_m = 100 \, \text{mA} = 0.1 \, A \quad \Rightarrow \quad I_{rms} = \frac{0.1}{\sqrt{2}}. \]
Step 2: Power factor.
The phase difference between $V$ and $i$ is $\phi = \frac{\pi}{3}$. \[ \cos \phi = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}. \]
Step 3: Average power consumed.
\[ P = V_{rms} \cdot I_{rms} \cdot \cos\phi = \frac{100}{\sqrt{2}} \cdot \frac{0.1}{\sqrt{2}} \cdot \frac{1}{2}. \] \[ P = \frac{100 \times 0.1}{2 \times 2} = \frac{10}{4} = 2.5 \, \text{W}. \]
Step 4: Conclusion.
The power consumed is 2.5 W.