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Chapter 30: Problem 12

In an RL circuit with alternating current, the current lags behind the voltage. What does this mean, and how can it be explained qualitatively, based on the phenomenon of electromagnetic induction?

Short Answer

Expert verified
Question: Explain the meaning of the statement "the current lags behind the voltage" in an RL circuit with alternating current, and qualitatively describe the role of electromagnetic induction in this phenomenon. Answer: In an RL circuit with alternating current, the statement "the current lags behind the voltage" means that the current waveform reaches its peak at a later time than the voltage waveform, with a phase difference between them. This occurs due to the phenomenon of electromagnetic induction. When current flows through the inductor, it creates a changing magnetic field, which induces an electromotive force (EMF) in the coil that opposes the change in current according to Lenz's law. The induced EMF is at its maximum when the rate of change of current is highest, which is when the voltage waveform reaches its peak. This results in the current waveform reaching its peak later, causing the current to lag behind the voltage.

Step by step solution

01

Understand RL circuits with alternating current

An RL circuit is an electrical circuit consisting of a resistor (R) and an inductor (L) connected in series with an alternating current (AC) voltage source. The voltage across the resistor and inductor varies sinusoidally with time. In such a circuit, the current also changes sinusoidally with time, but not necessarily in phase with the voltage. This means there may be a phase difference between the voltage and the current waveforms.
02

Analyze "current lags behind the voltage"

In an RL circuit, when we say the current lags behind the voltage, it means that the current wave reaches its peak at a later time than the voltage wave. In other words, the voltage waveform leads the current waveform, and thus, there is a phase difference between them. This phase difference is denoted by the symbol θ (in radians) or Φ (in degrees), and is given by the formula: θ = Φ = arctan (ωL/R) where ω is the angular frequency of the AC source, L is the inductance, and R is the resistance.
03

Relationship with electromagnetic induction

To qualitatively explain the "current lag" phenomenon in an RL circuit, we need to consider electromagnetic induction. When the AC voltage source drives the current in the circuit, the current flowing through the inductor (coil) creates a changing magnetic field around it. As per Faraday's law of electromagnetic induction, this changing magnetic field induces an electromotive force (EMF) in the coil, which opposes the change in current according to Lenz's law. In an RL circuit, the induced EMF is at its maximum when the rate of change of current is highest. This occurs when the voltage waveform reaches its peak, causing the maximum rate of change in the magnetic field and maximum opposition to the increase in the current. Consequently, the current waveform reaches its peak at a later time, resulting in the current lagging behind the voltage.
04

Conclusion

In summary, in an RL circuit with alternating current, the current lags behind the voltage due to the phenomenon of electromagnetic induction and the opposition provided by the induced EMF. As a result, there is a phase difference between the current and voltage waveforms, with the voltage waveform leading the current waveform. This phase difference is dependent on the circuit's inductance (L) and resistance (R) and can be calculated using the arctan (ωL/R) formula.

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Most popular questions from this chapter

30.24 A 2.00 - \(\mu\) F capacitor is fully charged by being connected to a 12.0 - \(\mathrm{V}\) battery. The fully charged capacitor is then connected to a \(0.250-\mathrm{H}\) inductor. Calculate (a) the maximum current in the inductor and (b) the frequency of oscillation of the LC circuit.

A \(200-\Omega\) resistor, a \(40.0-\mathrm{mH}\) inductor and a \(3.0-\mu \mathrm{F}\) capacitor are connected in series with a time-varying source of emf that provides \(10.0 \mathrm{~V}\) at a frequency of \(1000 \mathrm{~Hz}\). What is the impedance of the circuit? a) \(200 \Omega\) b) \(228 \Omega\) c) \(342 \Omega\) d) \(282 \Omega\)

An LC circuit consists of a capacitor, \(C=2.50 \mu \mathrm{F},\) and an inductor, \(L=4.0 \mathrm{mH}\). The capacitor is fully charged using a battery and then connected to the inductor. An oscilloscope is used to measure the frequency of the oscillations in the circuit. Next, the circuit is opened, and a resistor, \(R\), is inserted in series with the inductor and the capacitor. The capacitor is again fully charged using the same battery and then connected to the circuit. The angular frequency of the damped oscillations in the RLC circuit is found to be \(20 \%\) less than the angular frequency of the oscillations in the LC circuit. a) Determine the resistance of the resistor. b) How long after the capacitor is reconnected in the circuit will the amplitude of the damped current through the circuit be \(50 \%\) of the initial amplitude? c) How many complete damped oscillations will have occurred in that time?

An electromagnet consists of 200 loops and has a length of \(10.0 \mathrm{~cm}\) and a cross-sectional area of \(5.00 \mathrm{~cm}^{2}\). Find the resonant frequency of this electromagnet when it is attached to the Earth (treat the Earth as a spherical capacitor)

The figure shows a simple FM antenna circuit in which \(L=8.22 \mu \mathrm{H}\) and \(C\) is variable (the capacitor can be tuned to receive a specific station). The radio signal from your favorite FM station produces a sinusoidal time-varying emf with an amplitude of \(12.9 \mu \mathrm{V}\) and a frequency of \(88.7 \mathrm{MHz}\) in the antenna. a) To what value, \(C_{0}\), should you tune the capacitor in order to best receive this station? b) Another radio station's signal produces a sinusoidal time-varying emf with the same amplitude, \(12.9 \mu \mathrm{V}\), but with a frequency of \(88.5 \mathrm{MHz}\) in the antenna. With the circuit tuned to optimize reception at \(88.7 \mathrm{MHz}\), what should the value, \(R_{0}\), of the resistance be in order to reduce by a factor of 2 (compared to the current if the circuit were optimized for \(88.5 \mathrm{MHz}\) ) the current produced by the signal from this station?
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