A 2.0 - \(\mu\) F capacitor has \(1.0-\mathrm{k} \Omega\) reactance. (a) What's the frequency of the applied voltage? (b) What inductance would give the same reactance at this frequency? (c) How would the reactances compare if the frequency were doubled?

Capacitive reactance, symbolized as \(X_C\), is a measure of a capacitor's opposition to alternating current (AC). It varies inversely with both the frequency of the applied voltage and the capacitance value of the capacitor. Specifically, the capacitive reactance is calculated using the formula:
\[X_C = \frac{1}{2 \pi f C}\]
Here, \(f\) is the frequency in hertz (Hz) and \(C\) is the capacitance in farads (F). From this relationship, we can see that as frequency increases, the capacitive reactance decreases, which allows more current to pass through. Conversely, at lower frequencies, the reactance increases, inhibiting the flow of AC. This concept is crucial in tuning circuits and filters where specific frequencies must be blocked or passed through.

Contrary to capacitive components, inductive reactance, denoted as \(X_L\), increases with both the frequency and inductance. Inductors resist changes in current and the inductive reactance quantifies this opposition in an AC circuit. The formula to calculate inductive reactance is:
\[X_L = 2 \pi f L\]
where \(L\) is the inductance in henrys (H). Higher frequencies cause greater opposition to current flow through the inductor, leading to higher reactance. This is why inductors are used in conjunction with capacitors to manage frequency response in electronic devices.

Electrical resonance in AC circuits occurs when the capacitive and inductive reactances are equal in magnitude but opposite in phase, causing them to cancel each other out. This results in the overall impedance of the circuit being purely resistive, and the circuit naturally oscillates at this frequency, known as the resonant frequency. At resonance, a circuit can store and release energy, which can lead to voltage amplification. Resonance is the fundamental principle behind the operation of many electronic devices, including radios and televisions, where it is used to select desired frequencies from a wide spectrum of signals.

AC circuit analysis involves understanding how inductors and capacitors react to different frequencies of alternating current. These components are characterized by their reactance, which affects the total impedance in an AC circuit. Impedance, symbolized by \(Z\), is the combination of resistance and reactance (both capacitive and inductive) and is a measure of the overall opposition to current in an AC circuit.
Engineers use AC circuit analysis to design circuits with desired frequency characteristics, like achieving electrical resonance at a specific frequency. The calculations often involve solving complex equations that take into account the resistances, reactances, and the phase relationships between various components in the circuit.