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Inductive reactance is not a fixed value; it depends on the frequency of the alternating current passing through the inductor. This relationship is expressed mathematically as \( X_L = 2\pi f L \), where \( X_L \) is the inductive reactance, \( f \) is the frequency in hertz, and \( L \) is the inductance in henrys.

• Higher frequencies lead to higher reactance.
• Lower frequencies result in lower reactance.

This relationship shows that as the frequency changes, the reactance will adjust accordingly. In the given exercise, as the frequency decreases from 75.0 Hz to 60.0 Hz, the reactance also decreases, illustrating this dependency. Understanding this concept is crucial when calculating how different frequencies affect an inductor's behavior in AC circuits.

Finding the inductance of an inductor is a key step in analyzing its reactance at different frequencies. The formula \( L = \frac{X_L}{2\pi f} \) is derived from rearranging the standard reactance formula. For instance, with a given reactance of 56.5 Ω at a frequency of 75.0 Hz, substitute these values into the formula:\[L = \frac{56.5}{2\pi \times 75.0} \approx 0.12 \text{ H}\]This calculated value for inductance is a vital factor used to find the new reactance at a different frequency, ensuring that calculations are accurate and internally consistent. Inductance itself remains constant regardless of frequency, serving as a fundamental property of the inductor.

Manipulating the inductive reactance formula is essential for solving problems related to different frequencies and obtaining desired values such as inductance or reactance. Begin by understanding the basic form \( X_L = 2\pi f L \). To find the inductance given reactance and frequency, the formula can be rearranged into:\[L = \frac{X_L}{2\pi f}\]This step is crucial for calculating the inductance when initial conditions are known, as seen in the exercise. For calculating the new reactance at a different frequency:\[X_L = 2\pi f L\]Both forms demonstrate the versatility and necessity of formula manipulation for correctly analyzing circuits' responses to frequency changes. Having a firm grasp on this concept allows for solving a variety of electrical engineering problems efficiently.