The series combination of resistance R and inductance L is connected to an alternating source of e.m.f. `e = 311 sin (100 pit)`. If the value of wattless current is `0.5A` and the impedance of the circuit is `311Omega`, the power factor will be –

Options

A. `(1)/(2)`
B. `(1)/(sqrt2)`
C. `(1)/(sqrt3)`
D. `(1)/(sqrt5`

Explanation

To solve the problem step by step, we need to find the power factor of the circuit given the wattless current and the impedance.

### Step 1: Understand the given values
We have:
- Peak value of e.m.f. (E₀) = 311 V
- Wattless current (I_wattless) = 0.5 A
- Impedance (Z) = 311 Ω

### Step 2: Calculate the RMS value of the e.m.f.
The RMS (Root Mean Square) value of the e.m.f. can be calculated using the formula:
\[ E_{rms} = \frac{E_0}{\sqrt{2}} \]
Substituting the given value:
\[ E_{rms} = \frac{311}{\sqrt{2}} \]

### Step 3: Calculate the RMS current (I_rms)
Using the relationship between the RMS voltage, RMS current, and impedance:
\[ I_{rms} = \frac{E_{rms}}{Z} \]
Substituting the values:
\[ I_{rms} = \frac{311/\sqrt{2}}{311} = \frac{1}{\sqrt{2}} \, \text{A} \]

### Step 4: Relate wattless current to RMS current and sine of the phase angle
The wattless current is given by the formula:
\[ I_{wattless} = I_{rms} \sin \phi \]
Substituting the known values:
\[ 0.5 = \left(\frac{1}{\sqrt{2}}\right) \sin \phi \]

### Step 5: Solve for sin φ
Rearranging the equation:
\[ \sin \phi = 0.5 \sqrt{2} = \frac{\sqrt{2}}{2} \]

### Step 6: Calculate cos φ using the Pythagorean identity
Using the identity \( \sin^2 \phi + \cos^2 \phi = 1 \):
\[ \cos^2 \phi = 1 - \sin^2 \phi \]
Substituting the value of \( \sin \phi \):
\[ \cos^2 \phi = 1 - \left(\frac{\sqrt{2}}{2}\right)^2 = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 7: Calculate cos φ
Taking the square root:
\[ \cos \phi = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

### Step 8: Conclusion
The power factor (PF) is given by:
\[ \text{Power Factor} = \cos \phi = \frac{1}{\sqrt{2}} \]

### Final Answer
The power factor of the circuit is \( \frac{1}{\sqrt{2}} \).

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