91Ó°ÊÓ

Inductive reactance is a measure of an inductor's resistance to the flow of alternating current (AC). Unlike resistance in a resistor, which is constant, inductive reactance varies with the frequency of the AC signal. It is represented by the symbol \(X_L\) and is calculated using the formula \(X_L = 2\pi f L\), where \(f\) is the frequency and \(L\) is the inductance in henrys (H). High frequency AC signals will encounter greater inductive reactance, making inductors useful for filtering out unwanted high-frequency signals in electrical circuits.

To explain this concept in relation to our exercise, if we have a \(500\mu H\) inductor, at a certain frequency, the reactance \(X_L\) forms an opposition to the current, leading to lower current values as \(X_L\) increases. In the given exercise, calculating the reactance was the first step to determining the frequency at which the inductor will have a peak current of \(50\ mA\).

Analyzing an AC circuit involves understanding how inductors, capacitors, and resistors behave under an oscillating voltage supply. The voltages and currents in an AC circuit change over time, typically in a sinusoidal pattern, which means the values are not constant as they are in DC circuits. Key variables include the amplitude, frequency, and phase relationship between current and voltage.

Applying AC circuit analysis to our textbook exercise, we used the given peak voltage and desired peak current to find the reactance of the inductor. Further, we explored how the current's phase relationship to voltage affects the rate of change of current in the inductor, crucial for determining the emf at specific instances.

Ohm's law in the context of an inductor relates the voltage across an inductor to the time rate of change of the current through it, not just the current itself. The formula is \(V = L\cdot\frac{di}{dt}\), where \(V\) is the voltage across the inductor, \(L\) is the inductance, and \(\frac{di}{dt}\) is the rate of change of current. This relation is especially important since, in an inductor, the voltage leads the current by 90 degrees in phase, which implies that the maximum voltage occurs when the current is increasing or decreasing most rapidly -- not when the current is at its peak value.

For the given problem, when the rate of change of current (\(\frac{di}{dt}\)) is zero (at peak current), the induced emf as per Ohm's law for inductors is zero, which is counterintuitive compared to a resistor where the voltage peak corresponds to the current peak.

In alternating current (AC) systems, frequency is a crucial parameter, representing the number of cycles per second of the electrical signal, measured in hertz (Hz). The frequency determines how rapidly the voltage and current are oscillating. In inductive circuits, the frequency has a direct impact on reactance; as the frequency increases, the inductive reactance also increases, and vice versa.

In our exercise, finding the alternating current frequency at which the given inductor allows a peak current of \(50\ mA\) involved understanding and manipulating the reactance formula. The resulting frequency calculation required to solve the exercise gives us insight into the speed of oscillation necessary to produce the conditions described.