A capacitance C and an inductance L are operated at the same angular frequency. (a) At what angular frequency will they have the same reactance? (b) If L = 5 00 mH and C = 3.50 μF, what is the numerical value of the angular frequency in part (a), and what is the reactance of each element?

Verified step by step guidance

1. Understand that reactance is the opposition to the flow of alternating current caused by inductance and capacitance. For inductors, the reactance \( X_L \) is given by \( X_L = \omega L \), and for capacitors, the reactance \( X_C \) is given by \( X_C = \frac{1}{\omega C} \), where \( \omega \) is the angular frequency.
2. To find the angular frequency at which the reactance of the inductance and capacitance are equal, set \( X_L = X_C \). This gives the equation \( \omega L = \frac{1}{\omega C} \).
3. Solve the equation \( \omega L = \frac{1}{\omega C} \) for \( \omega \). Multiply both sides by \( \omega \) to get \( \omega^2 = \frac{1}{LC} \). Then, take the square root of both sides to find \( \omega = \frac{1}{\sqrt{LC}} \).
4. Substitute the given values \( L = 500 \text{ mH} = 0.500 \text{ H} \) and \( C = 3.50 \mu\text{F} = 3.50 \times 10^{-6} \text{ F} \) into the formula \( \omega = \frac{1}{\sqrt{LC}} \) to find the numerical value of the angular frequency.
5. Once you have the angular frequency, calculate the reactance of each element using \( X_L = \omega L \) and \( X_C = \frac{1}{\omega C} \) with the found \( \omega \).

Key Concepts

Reactance

Reactance is the resistance offered by capacitors and inductors to the flow of alternating current, due to their ability to store and release energy. Capacitive reactance (X_C) decreases with increasing frequency, while inductive reactance (X_L) increases. The reactance is crucial for determining how these components affect AC circuits.

Angular Frequency

Angular frequency, denoted by ω, is a measure of how quickly an AC circuit oscillates, expressed in radians per second. It is related to the frequency (f) by the formula ω = 2πf. Angular frequency is essential for calculating reactance and understanding the behavior of AC circuits at different frequencies.

Resonance in LC Circuits

Resonance occurs in LC circuits when the inductive reactance equals the capacitive reactance, leading to maximum energy transfer. At resonance, the angular frequency ω is given by ω = 1/√(LC). This concept is key to solving the problem, as it determines the frequency at which the reactances of the capacitor and inductor are equal.