A condenser of `10 mu F` and an inductor of `1 H` are connected in series with an `A.C` source of frequency `50 Hz`. The impedance of the combination will be (take `pi^(2) = 10)`

Options

A. zero
B. Inifinity
C. `44.7 Omega`
D. `5.67 Omega`

Explanation

To find the impedance of a series combination of a capacitor and an inductor connected to an AC source, we can follow these steps:

### Step 1: Identify the given values
- Capacitance, \( C = 10 \mu F = 10 \times 10^{-6} F \)
- Inductance, \( L = 1 H \)
- Frequency, \( f = 50 Hz \)
- Given \( \pi^2 = 10 \)

### Step 2: Calculate the angular frequency \( \omega \)
The angular frequency \( \omega \) is given by the formula:
\[ \omega = 2\pi f \]
Substituting the value of \( f \):
\[ \omega = 2 \pi \times 50 = 100 \pi \, \text{rad/s} \]

### Step 3: Calculate the inductive reactance \( X_L \)
The inductive reactance \( X_L \) is calculated using the formula:
\[ X_L = \omega L \]
Substituting the values:
\[ X_L = 100 \pi \times 1 = 100 \pi \, \Omega \]

### Step 4: Calculate the capacitive reactance \( X_C \)
The capacitive reactance \( X_C \) is calculated using the formula:
\[ X_C = \frac{1}{\omega C} \]
Substituting the values:
\[ X_C = \frac{1}{100 \pi \times 10 \times 10^{-6}} = \frac{1}{10^{-4} \pi} = \frac{10^4}{\pi} = \frac{10000}{\pi} \, \Omega \]

### Step 5: Calculate the impedance \( Z \)
Since there is no resistance in the circuit, the impedance \( Z \) is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
Since \( R = 0 \):
\[ Z = |X_L - X_C| \]
Substituting the values:
\[ Z = |100 \pi - \frac{10000}{\pi}| \]

### Step 6: Simplify the expression
To simplify, we find a common denominator:
\[ Z = \left| \frac{100 \pi^2 - 10000}{\pi} \right| \]
Substituting \( \pi^2 = 10 \):
\[ Z = \left| \frac{100 \times 10 - 10000}{\pi} \right| = \left| \frac{1000 - 10000}{\pi} \right| = \left| \frac{-9000}{\pi} \right| = \frac{9000}{\pi} \, \Omega \]

### Step 7: Final Calculation
Using \( \pi \approx 3.16 \):
\[ Z \approx \frac{9000}{3.16} \approx 2847.5 \, \Omega \]

### Conclusion
The impedance of the combination is approximately \( 2847.5 \, \Omega \).