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The concept of resonant frequency is crucial in understanding LRC circuits. It refers to the particular frequency at which the impedances of the inductor and capacitor, which have opposite reactive behaviors, cancel each other out. This cancellation leads to a circuit that is purely resistive, with zero reactive impedance, making the current and voltage in phase.

In mathematical terms, the resonant frequency \( f_0 \) is determined by the formula:\[\omega_0 = \frac{1}{\sqrt{LC}}\]where \( \omega_0 = 2\pi f_0 \), \( L \) is the inductance, and \( C \) is the capacitance. The resonant frequency depends solely on these two factors, indicating the ideal frequency for energy transfer in an LRC circuit.

At this frequency, the power factor hits its peak, and the circuit has minimum impedance. This is immensely useful in applications like tuning radios to the desired frequency or designing filters and oscillators, as it optimizes the circuit's performance.

In an LRC circuit, inductance \( L \) and capacitance \( C \) are key elements that determine the overall behavior of the circuit. Inductors and capacitors store energy, with inductors storing it in a magnetic field and capacitors in an electric field. Their relationship is fundamental to deriving the resonant frequency and understanding the dynamic behavior of the circuit.

**Key Points:**

• The impedance of an inductor \( Z_L = j \omega L \) is directly proportional to frequency (\( \omega \)).
• The impedance of a capacitor \( Z_C = \frac{-j}{\omega C} \) is inversely proportional to frequency.

When analyzing the condition \( V_L = 6V_C \) in the exercise, we equate current expressions to find:\[\omega L = 6 \times \left(\frac{1}{\omega C}\right)\]Solving this gives \( \omega^2 = \frac{6}{LC} \), illustrating how these two components interact in driving circuit behavior under certain conditions.

Angular frequency \( \omega \) represents how often an object rotates through a complete cycle. It is directly related to the frequency of the oscillations in the circuit, and in LRC circuits, it ties the resonant frequency with the normal operating frequency. This is because \( \omega = 2\pi f \), where \( f \) is the frequency in hertz (Hz), representing cycles per second.

In the context of the given problem, we explore how angular frequency defines the voltage relationship between components at a specified condition. We use the voltage ratio expression to derive:\[\omega^2 = 6 \times \omega_0^2\]Taking the square root, we find that \( \omega = \sqrt{6} \times \omega_0 \), showing that the angular frequency at which the condition \( V_L = 6V_C \) holds, is directly connected through a factor of \( \sqrt{6} \) to the resonant frequency.

This concept is crucial as it helps determine how a circuit behaves at different frequencies, allowing for strategic adjustments in real-world applications.