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Chapter 31: Problem 45

In an \(R L C\) circuit, can the amplitude of the voltage across an inductor be greater than the amplitude of the generator emf? (b) Consider an \(R L C\) circuit with \(\mathscr{E}_{m}=10 \mathrm{~V}, R=\) \(10 \Omega, L=1.0 \mathrm{H}\), and \(C=1.0 \mu \mathrm{F}\). Find the amplitude of the voltage across the inductor at resonance.

Short Answer

Expert verified
Yes, at resonance, the amplitude of the inductor voltage can exceed the generator emf. At resonance, the amplitude is 1000 V.

Step by step solution

01

Understanding the RLC Circuit

In an RLC circuit, the components (resistor, inductor, and capacitor) are connected in series with an AC generator. The generator emf (\( \mathscr{E}_m \)) drives the current through the circuit, creating voltage drops across each component.
02

Analyzing the Voltage Across the Inductor

The voltage across the inductor (\( V_L \)) can exceed the generator emf amplitude due to resonance. At resonance, the reactive impedance of the inductor (\( \omega L \)) and capacitor (\( \frac{1}{\omega C} \)) cancel each other out, allowing maximum current flow.
03

Calculating Resonant Frequency

The resonant frequency (\( \omega_0 \)) of an RLC circuit is given by:\[\omega_0 = \frac{1}{\sqrt{LC}}\]Substituting the given values, \( L = 1.0 \, \text{H} \) and \( C = 1.0 \, \mu \text{F} = 1.0 \times 10^{-6} \, \text{F} \), we find:\[\omega_0 = \frac{1}{\sqrt{1.0 \times 1.0 \times 10^{-6}}} = 1000 \, \text{rad/s}\]
04

Finding the Amplitude of the Voltage Across the Inductor

At resonance, \( V_L = L\omega_0I_m \). Where \( I_m \) is the maximum current, given by:\[ I_m = \frac{\mathscr{E}_m}{R} \]With \( \mathscr{E}_m = 10 \, \text{V} \) and \( R = 10 \, \Omega \):\[ I_m = \frac{10}{10} = 1 \, \text{A} \]Therefore, \[ V_L = 1.0 \, \text{H} \times 1000 \, \text{rad/s} \times 1 \, \text{A} = 1000 \, \text{V} \]This shows that the amplitude of the voltage across the inductor is 1000 V.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Resonance
Resonance is a fascinating phenomenon in RLC circuits that occurs under specific conditions. It happens when the inductive reactance and capacitive reactance in the circuit are equal and cancel each other out. This results in the circuit behaving as though it only consists of a resistor.
  • This cancellation allows the circuit to operate with maximum efficiency, letting the current reach its maximum value.
  • At resonance, the impedance of the circuit is minimized and equals just the resistance (R).
As there is no net reactive impedance to oppose the current, the voltage drops across the inductor and capacitor can become quite large, much more than the source voltage. This is why the inductor's voltage can exceed the generator's emf at resonance. The constructive interference of the voltages in the circuit enables this unique scenario.
Reactive Impedance
Reactive impedance is key to understanding how RLC circuits function, especially at resonance. Impedance (Z) is a measure of the total opposition that a circuit presents to the flow of alternating current (AC).
  • In an RLC circuit, this opposition comes from both resistance and reactance.
  • Reactance is the part of impedance that comes from the induction (inductive reactance) and capacitance (capacitive reactance) in the circuit.

Inductive reactance ( \( \omega L \) ) increases with frequency, while capacitive reactance ( \( \frac{1}{\omega C} \) ) decreases with frequency. At the resonant frequency, these reactances are equal but opposite, cancelling each other out. This cancellation explains why a circuit can have a very high current flow with minimal resistance, enabling the voltage across components like inductors to be much higher than anticipated.
Resonant Frequency
The resonant frequency ( \( \omega_0 \) ) is a fundamental concept that determines the conditions for resonance in an RLC circuit. This frequency is calculated using the expression:
\[ \omega_0 = \frac{1}{\sqrt{LC}} \]
This formula shows how the resonant frequency depends on both the inductance (L) and capacitance (C) of the circuit.
  • A high inductance or capacitance will lower the resonant frequency, leading the circuit to resonate at a lower frequency.
  • Conversely, lower inductance or capacitance will result in a higher resonant frequency.

When the circuit operates at this specific frequency, the reactance is balanced, and the impedance is at its minimum. This balance maximizes the current, allowing for larger voltage amplitudes across the inductor and capacitor, highlighting the profound effects of resonance in RLC circuits.

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Most popular questions from this chapter

A typical light dimmer used to dim the stage lights in a theater consists of a variable inductor \(L\) (whose inductance is adjustable between zero and \(\left.L_{\max }\right)\) connected in Fig. 31-33. The electrical supply is \(120 \mathrm{~V}\) (rms) at \(60.0 \mathrm{~Hz}\); the lightbulb is rated at \(120 \mathrm{~V}, 1000 \mathrm{~W}\). (a) What \(L_{\mathrm{max}}\) is required if the rate of energy dissipation in the lightbulb is to be varied by a factor of 5 from its upper limit of \(1000 \mathrm{~W}\) ? Assume that the resistance of the lightbulb is independent of its temperature. (b) Could one use a variable resistor (adjustable between zero and \(\left.R_{\max }\right)\) instead of an inductor? (c) If so, what \(R_{\max }\) is required? (d) Why isn't this done?

In an oscillating \(L C\) circuit, \(L=3.00 \mathrm{mH}\) and \(C=2.70\) \(\mu\) F. At \(t=0\) the charge on the capacitor is zero and the current is \(2.00\) A. (a) What is the maximum charge that will appear on the capacitor? (b) At what earliest time \(t>0\) is the rate at which energy is stored in the capacitor greatest, and (c) what is that greatest rate?

An ac generator provides emf to a resistive load in a remote factory over a two-cable transmission line. At the factory a step-down transformer reduces the voltage from its (rms) transmission value \(V_{t}\) to a much lower value that is safe and convenient for use in the factory. The transmission line resistance is \(0.30 \Omega /\) cable, and the power of the generator is 250 \(\mathrm{kW}\). If \(V_{t}=80 \mathrm{kV}\), what are (a) the voltage decrease \(\Delta V\) along the transmission line and (b) the rate \(P_{d}\) at which energy is dissipated in the line as thermal energy? If \(V_{t}=8.0 \mathrm{kV}\), what are (c) \(\Delta V\) and (d) \(P_{d}\) ? If \(V_{t}=0.80 \mathrm{kV}\), what are (e) \(\Delta V\) and (f) \(P_{d} ?\)

In an oscillating \(L C\) circuit, when \(75.0 \%\) of the total energy is stored in the inductor's magnetic field, (a) what multiple of the maximum charge is on the capacitor and (b) what multiple of the maximum current is in the inductor?

An oscillating \(L C\) circuit consisting of a \(1.0 \mathrm{nF}\) capacitor and a \(3.0 \mathrm{mH}\) coil has a maximum voltage of \(3.0 \mathrm{~V}\). What are (a) the maximum charge on the capacitor, (b) the maximum current through the circuit, and (c) the maximum energy stored in the magnetic field of the coil?
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