In a series L - C circuit, if `L = 10^(-3) H and C = 3xx10^(-7)` F is connected to a 100 V-50 Hz a.c.
Source, the impedance of the circuit is

Options

A. `(10^(5))/(3pi)-10pi`
B. `0.1pi -.3xx10^(-5)pi`
C. `(10^(5))/(3pi)-10pi`
D. None of these

Answer

To find the impedance of a series L-C circuit connected to an AC source, we can follow these steps:

### Step 1: Identify the given values
- Inductance, \( L = 10^{-3} \, \text{H} \)
- Capacitance, \( C = 3 \times 10^{-7} \, \text{F} \)
- Frequency, \( f = 50 \, \text{Hz} \)

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

### Step 3: Calculate the inductive reactance \( X_L \)
The inductive reactance \( X_L \) is given by:
\[ X_L = \omega L \]
Substituting the values of \( \omega \) and \( L \):
\[ X_L = 100 \pi \times 10^{-3} = 0.1 \pi \, \text{ohms} \]

### Step 4: Calculate the capacitive reactance \( X_C \)
The capacitive reactance \( X_C \) is given by:
\[ X_C = \frac{1}{\omega C} \]
Substituting the values of \( \omega \) and \( C \):
\[ X_C = \frac{1}{100 \pi \times 3 \times 10^{-7}} = \frac{1}{3 \times 10^{-5} \pi} = \frac{10^5}{3 \pi} \, \text{ohms} \]

### Step 5: Calculate the impedance \( Z \)
In a series L-C circuit, the impedance \( Z \) is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
Since the resistance \( R \) is not mentioned, we assume \( R = 0 \):
\[ Z = \sqrt{(X_L - X_C)^2} \]
Substituting the values of \( X_L \) and \( X_C \):
\[ Z = |X_L - X_C| = \left| 0.1 \pi - \frac{10^5}{3 \pi} \right| \]

### Step 6: Simplify the expression
To combine the terms, we can express \( 0.1 \pi \) in terms of a common denominator:
\[ 0.1 \pi = \frac{\pi}{10} \]
Thus, we have:
\[ Z = \left| \frac{\pi}{10} - \frac{10^5}{3 \pi} \right| \]

### Step 7: Final expression for impedance
The impedance can be expressed as:
\[ Z = \sqrt{\left( \frac{10^5}{3 \pi} - \frac{\pi}{10} \right)^2} \]
This gives us the final value of the impedance.

### Summary
The impedance \( Z \) of the series L-C circuit is:
\[ Z = \left| \frac{10^5}{3 \pi} - \frac{\pi}{10} \right| \]