In LCR circuit if `1/"LC" gt R^2/(4L^2)`, the circuit is

Options

A. oscillatory
B. dead beat
C. critically damped
D. none of the above

Answer

To solve the given problem, we need to analyze the conditions of an LCR circuit and determine the nature of the circuit based on the inequality provided.

### Step-by-Step Solution:

1. **Understanding the LCR Circuit**:
An LCR circuit consists of an inductor (L), a capacitor (C), and a resistor (R). The behavior of this circuit can be described using a second-order differential equation.

2. **Setting Up the Differential Equation**:
The general form of the differential equation for an LCR circuit is:
\[ L \frac{d^2I}{dt^2} + R \frac{dI}{dt} + \frac{1}{C} I = 0 \]
Here, \(I\) is the current through the circuit.

3. **Identifying the Characteristic Equation**:
The characteristic equation corresponding to the above differential equation is:
\[ Ls^2 + Rs + \frac{1}{C} = 0 \]
where \(s\) represents the complex frequency.

4. **Finding the Roots**:
The roots of this characteristic equation can be found using the quadratic formula:
\[ s = \frac{-R \pm \sqrt{R^2 - 4L \cdot \frac{1}{C}}}{2L} \]

5. **Analyzing the Discriminant**:
The discriminant \(D = R^2 - \frac{4L}{C}\) determines the nature of the roots:
- If \(D > 0\): Overdamped
- If \(D = 0\): Critically damped
- If \(D < 0\): Underdamped (oscillatory behavior)

6. **Applying the Given Condition**:
The problem states that:
\[ \frac{1}{LC} > \frac{R^2}{4L^2} \]
Rearranging this gives:
\[ \frac{R^2}{4L^2} < \frac{1}{LC} \]
This implies:
\[ R^2 < \frac{4L}{C} \]
Therefore, the discriminant becomes:
\[ R^2 - \frac{4L}{C} < 0 \]
This indicates that the roots are complex (imaginary), leading to an underdamped response.

7. **Conclusion**:
Since the condition \(1/LC > R^2/(4L^2)\) leads to an underdamped system, we conclude that the circuit is oscillatory.

### Final Answer:
The circuit is **oscillatory**.