In a R, L, C circult, three elements is connected in series by an ac source. If frequency is less than resonating frequency then net impedance of the circuit will be

Options

A. capacitive
B. inductive
C. capacitive or inductive
D. pure resistive

Answer

To solve the problem of finding the net impedance of an RLC circuit connected in series with an AC source when the frequency is less than the resonant frequency, we can follow these steps:

### Step 1: Understand the Components
In an RLC circuit, we have:
- R: Resistance
- L: Inductance
- C: Capacitance

The circuit is connected in series to an AC source.

### Step 2: Define Impedance
The net impedance (Z) of a series RLC circuit is given by the formula:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where:
- \( X_L \) is the inductive reactance, given by \( X_L = \omega L \)
- \( X_C \) is the capacitive reactance, given by \( X_C = \frac{1}{\omega C} \)

### Step 3: Understand Resonant Frequency
The resonant frequency (\( \omega_r \)) is defined as:
\[ \omega_r = \frac{1}{\sqrt{LC}} \]

### Step 4: Analyze the Given Condition
The problem states that the frequency (\( \omega \)) is less than the resonant frequency (\( \omega_r \)):
\[ \omega < \omega_r \]

### Step 5: Determine Reactances
Since \( \omega < \omega_r \):
- The inductive reactance \( X_L = \omega L \) will be smaller because \( \omega \) is less.
- The capacitive reactance \( X_C = \frac{1}{\omega C} \) will be larger because \( \omega \) is less.

Thus, we conclude:
\[ X_C > X_L \]

### Step 6: Calculate the Net Impedance
Since \( X_C > X_L \), we can express the difference as:
\[ X_C - X_L > 0 \]

Substituting this back into the impedance formula:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{R^2 + (-(X_C - X_L))^2} = \sqrt{R^2 + (X_C - X_L)^2} \]

This indicates that the circuit behaves more like a capacitive circuit.

### Step 7: Conclusion
When the frequency is less than the resonant frequency, the net impedance of the circuit will be dominated by the capacitive reactance, indicating that the circuit is capacitive in nature.

### Final Answer
The net impedance of the circuit will be:
- **Capacitive** (indicating that \( Z \) is influenced more by \( X_C \) than \( X_L \)).

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