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Chapter 28: Problem 6

When a particular inductor and capacitor are connected across the same AC voltage, the current in the inductor is higher than in the capacitor. Is this true at all frequencies?

Short Answer

Expert verified
No, the current in the inductor will not always be higher than in the capacitor at all frequencies. Depending on the frequency, either component could have a higher current.

Step by step solution

01

Understand Reactance

Reactance (X) is the opposition that inductors (\(X_L\)) and capacitors (\(X_C\)) present to current in AC circuits. It depends on the frequency of the alternating current (\(f\)) and the values of the inductance (\(L\)) and capacitance (\(C\)). The reactance of an inductor increases with frequency, while the reactance of a capacitor decreases with frequency. Mathematical expressions for reactance are: \(X_L= 2\pi f L\) for inductors and \(X_C= 1/(2\pi f C)\) for capacitors.
02

Analyze given condition

The problem states that when connected to the same AC voltage, the current in the inductor is greater than in the capacitor. In an AC circuit, the magnitude of current (\(I\)) is given by \(I=V/R\), where \(V\) is voltage and \(R\) is resistance. For an inductive or capacitive circuit, the reactance takes the place of resistance. Therefore, a higher current in the inductor than in the capacitor indicates that the inductive reactance (\(X_L\)) is lower than the capacitive reactance (\(X_C\)).
03

Apply concept to frequency

Since \(X_L= 2\pi f L\) and \(X_C= 1/(2\pi f C)\) the reactance of the inductor would be higher at higher frequencies, while the reactance of the capacitor would be lower. Thus, at low frequencies, \(X_L\) is smaller than \(X_C\), in accordance with the given situation. However, as the frequency increases, \(X_L\) will increase while \(X_C\) will decrease. Eventually, at some frequency (\(f\)), \(X_L\) will become equal to \(X_C\). Beyond this frequency, \(X_L\) will be greater than \(X_C\), reversing the situation. Thus, it's not true that the current in the inductor will always be higher than in the capacitor at all frequencies.

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Most popular questions from this chapter

You're Chief Financial Officer for a power company, and you consult your engineering department in an effort to minimize power-line losses. Your power plant produces \(60-\mathrm{Hz}\) power at \(365 \mathrm{kV} \mathrm{rms}\) and \(200 \mathrm{A} \mathrm{rms},\) and delivers it via transmission lines with total resistance \(100 \Omega .\) You ask the engineers for the percentage of power that's lost. They reply that it depends on the power factor. What's the percentage loss for power factors of (a) 1.0 and (b) \(0.60 ?\)

The same AC voltage appears across a capacitor and a resistor, and the same rms current flows in each. Is the power dissipation the same in each?

The voltage across two components in series is zero. Is it possible that the voltages across the individual components aren't zero? Give an example.

A series \(R L C\) circuit has \(R=18 \mathrm{k} \Omega, L=20 \mathrm{mH},\) and resonates at \(4.0 \mathrm{kHz}\). (a) What's the capacitance? (b) Find the circuit's impedance at resonance and (c) at \(3.0 \mathrm{kHz}\)

An LC circuit with \(C=18\) mF undergoes oscillations with period \(2.4 \mathrm{s}\). Find the inductance.
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