Inductors play a crucial role in alternating current (AC) circuits, influencing how voltage and current interact. In an AC circuit, the current oscillates over time, described by the equation I(t) = I_{max} \sin(\omega t), where I_{max} is the maximum current and \omega is the angular frequency. The voltage across an inductor is determined by Faraday's law, expressed as V_L = L \frac{dI}{dt}, where L is the inductance and \frac{dI}{dt} is the rate of change of current.

Using calculus, the voltage across an inductor in an AC circuit can be represented as V_L(t) = I_{max} \omega L \cos(\omega t + \frac{\pi}{2}). This indicates that the voltage across the inductor leads the current by 90 degrees, meaning that the voltage reaches its peak before the current does. In contrast, the voltage across a resistor is in phase with the current, represented by V_R(t) = I_{max} R \cos(\omega t). The phase difference between the voltage and current in an inductor is a key characteristic of inductive reactance.

The maximum voltage across an inductor can be calculated as V_{max} = I_{max} \omega L, where \omega L is known as the inductive reactance, analogous to resistance in resistors. The units for inductive reactance are ohms, similar to resistance.

To illustrate these concepts, consider an example where an AC power source provides a maximum voltage of 120 volts at a frequency of 60 hertz. If the maximum current in the circuit is 5 amps, we can find the inductance of the inductor. According to Kirchhoff's loop rule, the maximum current can be expressed as I_{max} = \frac{V_{max}}{X_L}, where X_L = \omega L. Rearranging gives us L = \frac{V_{max}}{\omega I_{max}}.

First, we calculate the angular frequency: \omega = 2\pi f = 2\pi \times 60 \text{ Hz} \approx 377 \text{ s}^{-1}. Substituting the known values into the inductance formula yields:

L = \frac{120 \text{ V}}{377 \text{ s}^{-1} \times 5 \text{ A}} \approx 0.064 \text{ H}.

This example highlights the relationship between voltage, current, and inductance in AC circuits, emphasizing the importance of understanding inductors in electrical engineering.

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