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In an AC circuit, capacitors exhibit a type of opposition to the flow of alternating current called capacitive reactance. It's similar to resistance but specific to AC circuits. Capacitive reactance, denoted as \( X_C \), is crucial in determining how much a capacitor will resist the AC signal passing through it. This reactance is frequency-dependent. The formula used to calculate capacitive reactance is:\[ X_C = \frac{1}{2\pi f C} \]where

• \( f \) is the frequency of the applied AC signal
• \( C \) is the capacitance in Farads

As the frequency increases, the capacitive reactance decreases. This means that at higher frequencies, capacitors allow more current to pass through. In the example, with a frequency of 50 Hz and a capacitance of \( 10^{-2} \) F, the capacitive reactance is calculated as \( \frac{1}{\pi} \) Ohms. You can see how even slight changes in frequency or capacitance can significantly affect the circuit's behavior.

Inductive reactance, denoted \( X_L \), is the property of an inductor in an AC circuit that opposes changes in current flow. Like capacitive reactance, it's also frequency-dependent but with an entirely different relationship to frequency. The formula for inductive reactance is:\[ X_L = 2\pi f L \]where

• \( f \) is the frequency of the AC source
• \( L \) is the inductance in Henrys

As frequency increases, so does the inductive reactance, meaning inductors will impede current flow more strongly at higher frequencies. In this exercise, with a frequency of 50 Hz and an inductance of 1 H, the inductive reactance is found to be \( 100\pi \) Ohms. This value shows how inductors can significantly influence the current in AC circuits, especially at varying frequencies.

A series LC circuit is a type of electric circuit that combines both a capacitor and an inductor connected in a series arrangement. The overall impedance of such a circuit is influenced by both the capacitive and inductive reactances. The impedance \( Z \) of the series LC circuit can be calculated with the formula:\[ Z = \sqrt{(X_L - X_C)^2} \]The difference \( X_L - X_C \) dictates whether the circuit behaves more like an inductor or a capacitor. If \( X_L > X_C \), the circuit has an inductive nature; if \( X_C > X_L \), it's more capacitive. In this task, the values derived were \( X_L = 100\pi \) and \( X_C = \frac{1}{\pi} \). The net impedance, therefore, was \( \left| 100\pi - \frac{1}{\pi} \right| \), demonstrating an inductive character since \( X_L > X_C \). This concept showcases the interplay between inductors and capacitors and how their reactances combine to affect overall circuit behavior.