A capacitor of `10 (mu)F` and an inductor of 1 H are joined in series. An ac of 50 Hz is applied to this combination. What is the impedance of the combination?

Options

A. `(5(pi^(2)-5))/(pi) Omega`
B. `(10(10-pi^(2)))/(pi) Omega`
C. `(10(pi^(2)-5))/(pi) Omega`
D. `(5(10-pi^(2)))/(pi) 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 \( f = 50 \, Hz \)

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

### Step 3: Calculate the capacitive reactance \( X_C \)
The capacitive reactance \( X_C \) is given by the formula:
\[ X_C = \frac{1}{\omega C} \]
Substituting the values of \( \omega \) and \( C \):
\[ X_C = \frac{1}{100 \pi \times 10 \times 10^{-6}} = \frac{1}{10^{-4} \pi} = \frac{10^4}{\pi} \, \Omega \]

### Step 4: Calculate the inductive reactance \( X_L \)
The inductive reactance \( X_L \) is given by the formula:
\[ X_L = \omega L \]
Substituting the values of \( \omega \) and \( L \):
\[ X_L = 100 \pi \times 1 = 100 \pi \, \Omega \]

### Step 5: Calculate the impedance \( Z \)
In a series circuit, the impedance \( Z \) is given by:
\[ Z = \sqrt{(R^2 + (X_L - X_C)^2)} \]
Since the resistance \( R \) is not given, we can assume \( R = 0 \):
\[ Z = |X_L - X_C| \]
Substituting the values of \( X_L \) and \( X_C \):
\[ Z = |100 \pi - \frac{10^4}{\pi}| \]

### Step 6: Calculate the numerical value
To find the numerical value, we can calculate:
1. \( 100 \pi \approx 314.16 \)
2. \( \frac{10^4}{\pi} \approx \frac{10000}{3.14} \approx 3183.1 \)

Now substituting back:
\[ Z = |314.16 - 3183.1| \approx 2868.94 \, \Omega \]

### Final Answer
Thus, the impedance of the combination is approximately:
\[ Z \approx 2868.94 \, \Omega \]