Frequency of AC source is continuously increased, impedance of LCR series circuit

Options

A. increases.
B. decreases.
C. remains constant.
D. first decreases to a minimum value and then increases.

Explanation

To solve the problem regarding the impedance of an LCR series circuit as the frequency of the AC source is continuously increased, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Impedance in LCR Circuit**:
The impedance (Z) of a series LCR circuit is given by the formula:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where:
- \( R \) is the resistance,
- \( X_L = 2\pi f L \) is the inductive reactance,
- \( X_C = \frac{1}{2\pi f C} \) is the capacitive reactance.

2. **Effect of Increasing Frequency**:
As the frequency \( f \) increases:
- The inductive reactance \( X_L \) increases because it is directly proportional to frequency (\( X_L = 2\pi f L \)).
- The capacitive reactance \( X_C \) decreases because it is inversely proportional to frequency (\( X_C = \frac{1}{2\pi f C} \)).

3. **Finding Resonance Condition**:
At a certain frequency, known as the resonant frequency \( f_0 \), the inductive reactance equals the capacitive reactance:
\[ X_L = X_C \implies 2\pi f_0 L = \frac{1}{2\pi f_0 C} \]
Solving for \( f_0 \) gives:
\[ f_0 = \frac{1}{2\pi \sqrt{LC}} \]

4. **Behavior of Impedance with Frequency**:
- As frequency increases from zero, \( X_C \) is initially larger than \( X_L \), making \( Z \) large.
- As frequency approaches \( f_0 \), \( X_L \) increases and \( X_C \) decreases, causing \( Z \) to decrease until it reaches a minimum at resonance.
- Beyond the resonant frequency, \( X_L \) becomes larger than \( X_C \), causing \( Z \) to increase again.

5. **Graphical Representation**:
The graph of impedance \( Z \) versus frequency \( f \) would show:
- A decrease in \( Z \) as \( f \) approaches \( f_0 \),
- A minimum point at \( f_0 \),
- An increase in \( Z \) as \( f \) continues to increase beyond \( f_0 \).

6. **Conclusion**:
Therefore, the impedance of the LCR series circuit first decreases to a minimum value at resonance and then increases as the frequency continues to increase.

### Final Answer:
The impedance of the LCR series circuit first decreases to a minimum value and then increases as the frequency of the AC source is continuously increased.