The r.m.s current in an `AC` circuit is `2A`. If the wattless current be `sqrt(3)A`, what is the power factor?

Options

A. `1/(sqrt(3))`
B. `1/(sqrt(2))`
C. `1/2`
D. `1/3`

Explanation

To solve the problem, we need to find the power factor of an AC circuit given the RMS current and the wattless current. Here’s a step-by-step solution:

### Step 1: Identify the given values
- RMS current (I_rms) = 2 A
- Wattless current (I_wattless) = √3 A

### Step 2: Understand the relationship between the components of the current
In an AC circuit, the total current can be broken down into two components:
1. The active current (I_active), which contributes to power.
2. The wattless current (I_wattless), which does not contribute to power.

The relationship can be expressed as:
- I_rms² = I_active² + I_wattless²

### Step 3: Express the active current in terms of the power factor
The active current can also be expressed using the power factor (pf):
- I_active = I_rms * cos(φ)

Where φ is the phase angle between the current and voltage.

### Step 4: Substitute the known values into the equation
Using the relationship:
- I_rms² = (I_rms * cos(φ))² + I_wattless²

Substituting the known values:
- (2 A)² = (2 A * cos(φ))² + (√3 A)²

### Step 5: Simplify the equation
- 4 = (2 * cos(φ))² + 3
- 4 = 4 * cos²(φ) + 3

### Step 6: Solve for cos²(φ)
Rearranging the equation gives:
- 4 * cos²(φ) = 4 - 3
- 4 * cos²(φ) = 1
- cos²(φ) = 1/4

### Step 7: Find cos(φ)
Taking the square root:
- cos(φ) = 1/2

### Step 8: Determine the power factor
The power factor (pf) is defined as:
- pf = cos(φ)

Thus, the power factor is:
- pf = 1/2

### Final Answer:
The power factor of the circuit is **1/2**.

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