A 50.0 mH inductor is connected as in Fig. 31-12 to an ac...

A 50.0 $\mathrm{mH}$ inductor is connected as in Fig. $31-12$ to an ac generator with $\mathscr{E}_{m}=30.0 \mathrm{V}$ . What is the amplitude of the resulting
alternating current if the frequency of the emf is (a) 1.00 $\mathrm{kHz}$ and
(b) 8.00 $\mathrm{kHz}$ ?

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Key Concepts

Inductive Reactance

Inductive reactance is the opposition offered by an inductor to the change in current in an AC circuit. It depends on both the frequency of the alternating current and the inductance of the coil, and is given by the formula X_L = ?L, where ? is the angular frequency. This concept is crucial for analyzing circuits containing inductors, as it determines how the inductor limits the current flow at different frequencies.

Angular Frequency

Angular frequency is a measure that relates the regular frequency of an AC source to the rate of oscillation in radians per second, defined by the formula ? = 2?f. It converts the frequency from cycles per second to radians per second, which is necessary when using trigonometric relationships and calculating reactance in AC circuits.

AC Ohm’s Law

AC Ohm’s Law extends the traditional Ohm’s Law to alternating current circuits by including the concept of impedance, which combines resistance and reactance. In circuits with only inductive elements, the impedance is purely reactive (X_L), and the amplitude of the current can be found by dividing the amplitude of the voltage by the inductive reactance. This approach provides a method to calculate current in circuits where energy storage rather than energy dissipation is predominant.

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