A 120-V ac voltage source is connected across a pure 0.700-H inductor. Find the current through

step 1 we have to find the value of first current through the inductor for 60 huts and 60 kilohertz and

A 120-V ac voltage source is connected across a pure 0.700-H...

A $120-\mathrm{V}$ ac voltage source is connected across a pure $0.700-\mathrm{H}$ inductor. Find the current through the inductor if the frequency of the source is $(a) 60.0 \mathrm{~Hz}$ and $(b) 60.0 \mathrm{kHz} .(c)$ What is the power loss in the inductor? $$ \text { (a) } X_{L}=2 \pi f L=2 \pi\left(60.0 \mathrm{~s}^{-1}\right)(0.700 \mathrm{H})=264 \Omega $$ Then $$ I=\frac{V}{X_{L}}=\frac{120 \mathrm{~V}}{264 \Omega}=0.455 \mathrm{~A} $$ (b) Now $X_{L}=264 \times 10^{3} \Omega$, so $I=0.455 \times 10^{-3} \mathrm{~A} .$ Notice that the impedance of an inductor varies directly with the frequency. (c) Inasmuch as $\cos \phi=R / Z$ and $R=0$; $$ \text { Power loss }=V I \cos \phi=V I \cos 90^{\circ}=0 $$

A $120-\mathrm{V}$ ac voltage source is connected across a pure $0.700-\mathrm{H}$ inductor. Find the current through the inductor if the frequency of the source is $(a) 60.0 \mathrm{~Hz}$ and $(b) 60.0 \mathrm{kHz} .(c)$ What is the power loss in the inductor?
$$
\text { (a) } X_{L}=2 \pi f L=2 \pi\left(60.0 \mathrm{~s}^{-1}\right)(0.700 \mathrm{H})=264 \Omega
$$
Then
$$
I=\frac{V}{X_{L}}=\frac{120 \mathrm{~V}}{264 \Omega}=0.455 \mathrm{~A}
$$
(b) Now $X_{L}=264 \times 10^{3} \Omega$, so $I=0.455 \times 10^{-3} \mathrm{~A} .$ Notice that the impedance of an inductor varies directly with the frequency.
(c) Inasmuch as $\cos \phi=R / Z$ and $R=0$;
$$
\text { Power loss }=V I \cos \phi=V I \cos 90^{\circ}=0
$$

Key Concepts

Inductive Reactance

Inductive reactance is the opposition that an inductor presents to a change in current in an AC circuit, and it is dependent on both the frequency of the source and the inductance value. It is calculated using the formula XL = 2?fL, which shows that the reactance increases linearly with frequency. This makes the inductor behave differently at varying frequencies, providing low impedance at low frequencies and high impedance at high frequencies.

AC Circuit Analysis

AC circuit analysis involves examining circuits powered by alternating current sources, where voltage and current vary sinusoidally with time. Key components such as resistors, inductors, and capacitors are analyzed using their impedances, which introduce magnitude and phase considerations. Understanding the behavior of these components under AC conditions is essential for predicting how voltage, current, and power are distributed throughout the circuit.

Frequency Dependence

The dependence of an inductor's reactance on frequency is a key concept in AC circuits. Since inductive reactance is proportional to frequency, altering the frequency of the AC source changes the inductor's impedance significantly. This concept is crucial when designing circuits that must operate over a range of frequencies or require filtering specific frequency components.

Phase Difference and Power Loss in Inductors

In an ideal inductor, the current lags the voltage by 90 degrees, leading to a situation where the power factor is zero. This phase difference means that the reactive power oscillates back and forth between the source and the inductor without being dissipated as real power, resulting in no net power loss. This is a distinguishing feature in pure reactive components compared to resistive components, where energy is actually dissipated.

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