In an AC circuit V and I are given by `V = 100 sin 100t V and I = 100 sin (100t + pi//3)` mA. The power dissipated in the circuit is

Options

A. `10_(4) watt`
B. 10 watt
C. 2.5 watt
D. 5 watt

Answer

To solve the problem of finding the power dissipated in the given AC circuit, we can follow these steps:

### Step 1: Identify the given values
The voltage \( V \) and current \( I \) are given as:
- \( V = 100 \sin(100t) \) volts
- \( I = 100 \sin(100t + \frac{\pi}{3}) \) mA

### Step 2: Determine the peak values
From the equations:
- The peak voltage \( V_0 = 100 \) V
- The peak current \( I_0 = 100 \) mA

### Step 3: Calculate the RMS values
The root mean square (RMS) values are calculated using the formula:
\[ V_{rms} = \frac{V_0}{\sqrt{2}} \quad \text{and} \quad I_{rms} = \frac{I_0}{\sqrt{2}} \]
Substituting the peak values:
\[ V_{rms} = \frac{100}{\sqrt{2}} \quad \text{and} \quad I_{rms} = \frac{100}{\sqrt{2}} \text{ mA} \]

### Step 4: Convert current to amperes
Since \( I_{rms} \) is in mA, we convert it to amperes:
\[ I_{rms} = \frac{100}{\sqrt{2}} \times 10^{-3} \text{ A} \]

### Step 5: Determine the phase angle
The phase angle \( \phi \) is given as \( \frac{\pi}{3} \).

### Step 6: Calculate the power factor
The power factor \( \cos(\phi) \) is:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

### Step 7: Calculate the power dissipated
The power \( P \) dissipated in the circuit can be calculated using the formula:
\[ P = V_{rms} \times I_{rms} \times \cos(\phi) \]
Substituting the values:
\[ P = \left(\frac{100}{\sqrt{2}}\right) \times \left(\frac{100 \times 10^{-3}}{\sqrt{2}}\right) \times \frac{1}{2} \]
\[ = \frac{100 \times 100 \times 10^{-3}}{2} \times \frac{1}{2} \]
\[ = \frac{10000 \times 10^{-3}}{4} = \frac{10}{4} = 2.5 \text{ W} \]

### Final Answer
The power dissipated in the circuit is \( 2.5 \text{ W} \).

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