Example 28.2: A 0.470-μF capacitor and a 144-μH inductor are connected across identical AC volta

step 1 In this problem, we're told we have a capacitor and an inductor, and we want to know if they are

Example 28.2: A 0.470-μF capacitor and a 144-μH inductor are...

Example 28.2: A $0.470-\mu \mathrm{F}$ capacitor and a $144-\mu \mathrm{H}$ inductor are connected across identical AC voltage sources. (a) If the rms current in the capacitor is the same as in the inductor, what's the frequency of the sources?

Key Concepts

Capacitive Reactance

Capacitive reactance (X?C?) is the measure of a capacitor's opposition to changes in voltage in an AC circuit. It is defined as X?C? = 1/(?C), where ? is the angular frequency and C is the capacitance. This concept is key in AC analysis because it determines how much current will flow through the capacitor relative to an applied voltage at a given frequency.

Inductive Reactance

Inductive reactance (X?L?) quantifies the opposition an inductor offers to changes in current in an AC circuit. It is given by X?L? = ?L, where ? is the angular frequency and L is the inductance. Understanding inductive reactance is essential for analyzing the behavior of circuits containing inductors, especially when comparing them to capacitive effects.

Angular Frequency and Frequency

Angular frequency (?) and frequency (f) are fundamental concepts in AC circuit analysis. Angular frequency is defined as ? = 2?f, connecting the familiar frequency in hertz (cycles per second) to the radians per second measurement used in reactance formulas. This conversion plays a critical role in determining the behavior of reactive components at different AC frequencies.

Resonance Condition in LC Circuits

The resonance condition emerges in LC circuits when the magnitudes of the capacitive and inductive reactances are equal, leading to the equality 1/(?C) = ?L. At this resonant frequency, the energy oscillates between the capacitor and inductor, and the overall circuit exhibits unique behavior such as maximum current flow. This concept is central to understanding how the frequency can be determined when the reactive effects of a capacitor and an inductor are set to be equivalent.

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