Suppose an inductor is connected directly to an AC voltage source, as shown in Figure 1. It is reasonable to assume negligible resistance, since in practice we can make the resistance of an inductor so small that it has a negligible effect on the circuit. Also shown is a graph of voltage and current as functions of time.

The graph in Figure 1(b) starts with voltage at a maximum. Note that the current starts at zero and rises to its peak *after* the voltage that drives it, just as was the case when DC voltage was switched on in the preceding section. When the voltage becomes negative at point a, the current begins to decrease; it becomes zero at point b, where voltage is its most negative. The current then becomes negative, again following the voltage. The voltage becomes positive at point c and begins to make the current less negative. At point d, the current goes through zero just as the voltage reaches its positive peak to start another cycle. This behavior is summarized as follows:

**AC Voltage in an Inductor: **

When a sinusoidal voltage is applied to an inductor, the voltage leads the current by one-fourth of a cycle, or by a

Current lags behind voltage, since inductors oppose change in current. Changing current induces a back emf

where

with *,* small

### Example 1: **Calculating Inductive Reactance and then Current**

(a) Calculate the inductive reactance of a 3.00 mH inductor when 60.0 Hz and 10.0 kHz AC voltages are applied. (b) What is the rms current at each frequency if the applied rms voltage is 120 V?

*Strategy*

The inductive reactance is found directly from the expression

*Solution for (a)*

Entering the frequency and inductance into Equation

Similarly, at 10 kHz,

*Solution for (b)*

The rms current is now found using the version of Ohm’s law in Equation

Similarly, at 10 kHz,

*Discussion*

The inductor reacts very differently at the two different frequencies. At the higher frequency, its reactance is large and the current is small, consistent with how an inductor impedes rapid change. Thus high frequencies are impeded the most. Inductors can be used to filter out high frequencies; for example, a large inductor can be put in series with a sound reproduction system or in series with your home computer to reduce high-frequency sound output from your speakers or high-frequency power spikes into your computer.

Note that although the resistance in the circuit considered is negligible, the AC current is not extremely large because inductive reactance impedes its flow. With AC, there is no time for the current to become extremely large.

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