Question

The voltage and current of an ac circuit are represented as \( V = 100 \sin(100t) \) volt and \( i = 100 \sin(100t + \frac{\pi}{3}) \) mA respectively. The power dissipated in the circuit is:

• A. \( 10^4 \) watt
• B. 2.5 watt
• C. 0.25 watt
• D. 25 watt

Hint:

The power dissipated in an AC circuit is given by the product of the maximum voltage, maximum current, and the cosine of the phase difference between them.

Answer 1

The Correct Option is B

The instantaneous power dissipated in an AC circuit is given by the formula:
\[ P = V_{\text{max}} I_{\text{max}} \cos(\phi) \] Where:
- \( V_{\text{max}} \) is the maximum voltage,
- \( I_{\text{max}} \) is the maximum current,
- \( \phi \) is the phase difference between the voltage and current.
From the given expressions:
- \( V = 100 \sin(100t) \), so \( V_{\text{max}} = 100 \) volts,
- \( i = 100 \sin(100t + \frac{\pi}{3}) \), so \( I_{\text{max}} = 100 \) mA or \( 0.1 \) A,
- The phase difference \( \phi = \frac{\pi}{3} \).
Now, substitute these values into the power formula:
\[ P = (100)(0.1) \cos\left(\frac{\pi}{3}\right) \] Since \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \), we get:
\[ P = (100)(0.1) \times \frac{1}{2} = 2.5 \, \text{watt} \] Thus, the correct answer is option (B) 2.5 watt.