A resistor and capacitor are connected to an ac supply of 200 V, 50 Hz in series. The current in the circuit is 2 A. If the power consumed in the circuit is 100 W then in the above question, the capacitive reactance in the circuit is

Options

A. `100 Omega`
B. `25 Omega`
C. `sqrt(125xx75)Omega`
D. `400 Omega`

Explanation

To find the capacitive reactance (X_C) in the given AC circuit with a resistor and capacitor, we can follow these steps:

### Step 1: Identify the given values
- Voltage (V_RMS) = 200 V
- Frequency (f) = 50 Hz
- Current (I_RMS) = 2 A
- Power (P) = 100 W

### Step 2: Calculate the resistance (R)
Using the formula for power in an AC circuit:
\[ P = I_{RMS}^2 \times R \]
We can rearrange this to find R:
\[ R = \frac{P}{I_{RMS}^2} \]
Substituting the known values:
\[ R = \frac{100 \text{ W}}{(2 \text{ A})^2} = \frac{100}{4} = 25 \, \Omega \]

### Step 3: Calculate the impedance (Z)
The impedance in the circuit can be calculated using Ohm's law:
\[ Z = \frac{V_{RMS}}{I_{RMS}} \]
Substituting the known values:
\[ Z = \frac{200 \text{ V}}{2 \text{ A}} = 100 \, \Omega \]

### Step 4: Relate impedance, resistance, and capacitive reactance
In an R-C series circuit, the relationship between impedance, resistance, and capacitive reactance is given by:
\[ Z^2 = R^2 + X_C^2 \]
Substituting the values we have:
\[ 100^2 = 25^2 + X_C^2 \]
This simplifies to:
\[ 10000 = 625 + X_C^2 \]

### Step 5: Solve for capacitive reactance (X_C)
Rearranging the equation to isolate X_C:
\[ X_C^2 = 10000 - 625 \]
\[ X_C^2 = 9375 \]
Taking the square root gives:
\[ X_C = \sqrt{9375} \]

### Step 6: Simplify the square root
We can simplify \( \sqrt{9375} \):
\[ 9375 = 125 \times 75 \]
Thus:
\[ X_C = \sqrt{125 \times 75} = \sqrt{125} \times \sqrt{75} \]
Calculating these values:
\[ \sqrt{125} = 5\sqrt{5} \]
\[ \sqrt{75} = 5\sqrt{3} \]
So:
\[ X_C = 5\sqrt{5} \times 5\sqrt{3} = 25\sqrt{15} \]

### Final Answer
Thus, the capacitive reactance in the circuit is:
\[ X_C \approx 96.2 \, \Omega \] (approximately, depending on the simplification)