What is the reactance of a \(10-\mu \mathrm{F}\) capacitor at a frequency of \(1 \mathrm{kHz}\) ? For networks in which the resistor levels are typically in the kilohm range, is it a good assumption to use the short-circuit equivalence for the conditions just described? How about at \(100 \mathrm{kHz}\) ?

When analyzing an Alternating Current (AC) circuit, one must understand that the voltage and current change over time. Unlike Direct Current (DC) circuits where current flows uniformly, AC circuits are characterized by their frequency, which indicates how many times per second the current changes direction. For AC circuits, components like capacitors and inductors have a property called reactance that impedes the flow of current depending on the frequency of the supply voltage.

Capacitors, in particular, have a reactance that decreases as the frequency increases, which means they readily allow higher frequency signals to pass while impeding low-frequency signals. This nature of capacitors to resist changes in voltage makes them crucial for AC circuit analysis, as it determines how they will interact with other components like resistors and inductors in a circuit.

In the realm of AC circuit analysis, impedance plays a central role. It is the combined effect of resistance, inductive reactance, and capacitive reactance. Resistance is frequency-independent, but reactance is frequency-dependent. The total impedance in a circuit is a complex quantity, comprising both resistance (R) and reactance (X), which can be inductive (XL) or capacitive (XC).

The formula for capacitive reactance is \(X_{C} = \frac{1}{2\pi f C}\), where \(f\) is the frequency and \(C\) is the capacitance. As frequency increases, capacitive reactance decreases. This unique relationship affects how AC circuits are analyzed because the impedance at different frequencies will determine the circuit's behavior. For example, at higher frequencies, a capacitor may behave much like a short circuit because its reactance is low, allowing current to pass through easily.

The frequency response of capacitors is a critical concept in understanding how they affect AC circuit behavior. A capacitor’s ability to store and release energy varies with frequency. At low frequencies, a capacitor has high reactance, acting almost like an open circuit and blocking the flow of current. As the frequency increases, its reactance decreases, allowing more current to pass through.

In the exercise, it’s examined whether the reactance level allows for treating the capacitor as a short-circuit at certain frequencies. This simplification is based on whether the reactance is significantly less than typical resistance levels in the circuit. For instance, at 1 kHz, the capacitor has a certain reactance. When the frequency is increased to 100 kHz, its reactance drops drastically, potentially justifying the use of a short-circuit model for the capacitor at this higher frequency. Understanding this behavior helps in designing and analyzing circuits, especially in filtering applications where certain frequency components need to be blocked or allowed to pass.