In an electric circuit the applied alternating emf is given by E = 100 sin (314 t) volt, and current flowing `I = sin (314t + pi//3)`. Then the impedance of the circuit is (in ohm) -

Options

A. `100 // sqrt(2)`
B. 100
C. `100 sqrt(2)`
D. None of the above

Answer

To find the impedance of the circuit given the alternating emf and the current, we can follow these steps:

### Step 1: Identify the given values
- The applied alternating emf is given by:
\[ E = 100 \sin(314t) \text{ volts} \]
Here, the peak voltage \( E_0 = 100 \) volts and the angular frequency \( \omega = 314 \) rad/s.

- The current flowing in the circuit is given by:
\[ I = \sin(314t + \frac{\pi}{3}) \]
This indicates that the peak current \( I_0 = 1 \) ampere and there is a phase difference of \( \frac{\pi}{3} \) radians between the current and the voltage.

### Step 2: Calculate the RMS values
- The root mean square (RMS) values for voltage and current can be calculated as follows:
\[ V_{\text{rms}} = \frac{E_0}{\sqrt{2}} = \frac{100}{\sqrt{2}} = 50\sqrt{2} \text{ volts} \]
\[ I_{\text{rms}} = \frac{I_0}{\sqrt{2}} = \frac{1}{\sqrt{2}} \text{ amperes} \]

### Step 3: Use the relationship between voltage, current, and impedance
- The relationship between the RMS voltage, RMS current, and impedance \( Z \) is given by:
\[ Z = \frac{V_{\text{rms}}}{I_{\text{rms}}} \]

### Step 4: Substitute the RMS values into the impedance formula
- Substitute the values calculated:
\[ Z = \frac{50\sqrt{2}}{\frac{1}{\sqrt{2}}} \]
Simplifying this gives:
\[ Z = 50\sqrt{2} \cdot \sqrt{2} = 50 \cdot 2 = 100 \text{ ohms} \]

### Conclusion
- The impedance of the circuit is:
\[ Z = 100 \text{ ohms} \]

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