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Inductance is a fundamental property of electrical circuits that reflects the ability of a component, typically an inductor, to store energy in the form of a magnetic field. In the exercise, the inductor's value was given as 30.0 millihenrys (mH), which is equal to 0.03 henrys (H) when converted.

Understanding inductance is crucial in AC circuit analysis because it determines how an inductor will react to changing currents. An inductor with high inductance will store more energy and thus have a higher opposition to the change in current. These properties affect how the inductor responds at different frequencies, a concept essential to designing circuits like filters and oscillators.

AC circuit analysis involves understanding how alternating current (AC) flows through various circuit components, like resistors, capacitors, and inductors. An AC current changes direction periodically and is characterized mainly by its frequency—the number of cycles it completes in one second, measured in hertz (Hz).

When analyzing an inductor in an AC circuit, its reactance becomes a key factor. Reactance is the opposition that inductors (and capacitors) exhibit to the change in current. Unlike resistors, which have resistance that remains constant irrespective of frequency, the reactance of inductors changes with frequency. Higher frequencies result in higher inductive reactance. In the given exercise, we examine how the frequency affects the inductor's reactance.

The reactance formula for an inductor is expressed as \(X_L = 2 \pi f L\), where \(X_L\) is the inductive reactance in ohms (\(\Omega\)), \(f\) is the frequency in hertz (Hz), and \(L\) is the inductance in henrys (H). This formula shows that the inductive reactance is directly proportional to both the frequency and the inductance.

To find the frequency that yields a specific reactance, as we do in the exercise, the formula can be rearranged: \(f = \frac{X_L}{2 \pi L}\). This equation indicates that as the frequency increases, the reactance also increases. By substituting the known reactance and inductance into this formula, we can solve for the unknown frequency. This is a powerful tool in AC circuit analysis, allowing designers to predict how an inductor will behave in a circuit at different frequencies.