The potential differences across the resistance, capacitance and inductance are `80 V, 40 V` and `100 V` respectively in an `L-C-R` circuit. The power factor of this circuit is

Options

A. 0.4
B. 0.5
C. 0.8
D. 1

Explanation

To find the power factor of the given L-C-R circuit, we will follow these steps:

### Step 1: Identify the given values
The potential differences across the components are:
- Voltage across the resistor (V_R) = 80 V
- Voltage across the capacitor (V_C) = 40 V
- Voltage across the inductor (V_L) = 100 V

### Step 2: Calculate the impedance (Z)
The impedance (Z) in an L-C-R circuit can be calculated using the formula:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where:
- \( R \) is the resistance,
- \( X_L \) is the inductive reactance,
- \( X_C \) is the capacitive reactance.

From the given voltages, we can identify:
- \( V_R = I \cdot R \) (where \( I \) is the current)
- \( V_L = I \cdot X_L \)
- \( V_C = I \cdot X_C \)

### Step 3: Express the reactances in terms of current
From the above relationships, we can express the reactances as:
- \( R = \frac{V_R}{I} = \frac{80}{I} \)
- \( X_L = \frac{V_L}{I} = \frac{100}{I} \)
- \( X_C = \frac{V_C}{I} = \frac{40}{I} \)

### Step 4: Substitute into the impedance formula
Substituting the expressions for \( R \), \( X_L \), and \( X_C \) into the impedance formula:
\[ Z = \sqrt{\left(\frac{80}{I}\right)^2 + \left(\frac{100}{I} - \frac{40}{I}\right)^2} \]
This simplifies to:
\[ Z = \sqrt{\left(\frac{80}{I}\right)^2 + \left(\frac{60}{I}\right)^2} \]
\[ Z = \sqrt{\frac{6400}{I^2} + \frac{3600}{I^2}} = \sqrt{\frac{10000}{I^2}} = \frac{100}{I} \]

### Step 5: Calculate the power factor (cos φ)
The power factor (cos φ) is given by:
\[ \text{Power Factor} = \frac{R}{Z} \]
Substituting the values we found:
\[ \text{Power Factor} = \frac{\frac{80}{I}}{\frac{100}{I}} = \frac{80}{100} = 0.8 \]

### Final Answer
The power factor of the circuit is **0.8**.

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