91Ó°ÊÓ

(a) In a series LC circuit driven by a DC voltage \((\omega=0)\), compare the energy stored in the inductor to the energy stored in the capacitor. (b) Carry out the same comparison for an LC circuit that is oscillating freely (without any driving voltage). (c) Now consider the general case of a series LC circuit driven by an oscillating voltage at an arbitrary frequency. Let \(\overline{U_{L}}\) and be the average energy stored in the inductor, and similarly for \(\overline{U_{C}}\). Define a quantity \(u=\overline{U_{C}} /\left(\overline{U_{L}}+\overline{U_{C}}\right)\), which can be interpreted as the capacitor's average share of the energy, while \(1-u\) is the inductor's average share. Find \(u\) in terms of \(L, C\), and \(\omega\), and sketch a graph of \(u\) and \(1-u\) versus \(\omega\). What happens at resonance? Make sure your result is consistent with your answer to part a.(answer check available at lightandmatter.com)

Step by step solution

01

Analyze the Energy in a DC-driven LC Circuit

When a DC voltage is applied (d0), the inductor acts as a short circuit after some time due to constant current, resulting in zero voltage across it. Therefore, energy stored in the inductor is zero. The capacitor stores energy given by \( \frac{1}{2} C V^2 \). Since the inductor stores no energy, the energy in the capacitor is predominant.

02

Analyze Energy in a Freely Oscillating LC Circuit

In an LC circuit without driving voltage, energy oscillates between the inductor and the capacitor. When energy is fully stored in the capacitor, it is \( \frac{1}{2} C V^2 \), and when fully in the inductor, it is \( \frac{1}{2} L I^2 \). This energy exchange follows harmonic motion, and total energy remains constant, with equal average energy over time in both components.

03

General Case with an Oscillating Voltage Source

For a driven LC circuit at frequency \( \omega \), energies stored in the components are defined by their capacitive and inductive reactances. The average energy in the capacitor is \( \overline{U_{C}} = \frac{1}{2} C |V_{max}|^2 \), and in the inductor is \( \overline{U_{L}} = \frac{1}{2} L |I_{max}|^2 \). At resonance \( \omega = \frac{1}{\sqrt{LC}} \), reactances balance out, and energy distribution is equal. Therefore, \( u = \frac{\overline{U_{C}}}{\overline{U_{C}} + \overline{U_{L}}} = \frac{1}{2} \).

04

Deriving the Expression for u

Using the formulas for reactance, at a general frequency \( \omega \), \( |I_{max}| = \omega C |V_{max}| \) and \( |U_{max|I} = \frac{|V_{max}|}{\omega L} \). Integrating energy equations gives \( u = \frac{1}{1 + \left(\frac{c}{L\omega^2}\right)} \). As \( \omega \to \omega_0 \), where \( \omega_0 \) is the natural frequency, \(u\) tends to 0.5. This indicates balanced energy share at resonance.

05

Graphing the Results

Graph \( u \) and \( 1-u \) on the y-axis against \( \omega \) on the x-axis. At low frequencies (\( \omega = 0 \)), \( u \) approaches 1. At resonance, both \( u \) and \( 1-u \) equal 0.5. At high frequencies, \( u \rightarrow 0 \) because inductor's reactance predominates. This graph shows how the energy distribution shifts with frequency, consistent with previous steps.

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Key Concepts

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

Capacitor Energy Storage

In an LC circuit, the capacitor plays a key role in energy storage. When connected to a DC supply, the capacitor stores energy as an electric field between its plates. At this point, the formula used to express the energy stored in the capacitor is \( \frac{1}{2} C V^2 \), where \( C \) is the capacitance and \( V \) is the voltage across the capacitor.
The energy in the capacitor peaks when the entire supplied voltage is across it, as no current flows through the inductor. This configuration ensures that while the inductor stores no energy, the capacitor holds all the system's energy.

Inductor Energy Storage

In contrast to the capacitor, the inductor stores energy in the form of a magnetic field when current flows through it. In a freely oscillating LC circuit, energy transitions between the capacitor and the inductor without external power sources.
At its peak, the energy in the inductor can be expressed as \( \frac{1}{2} L I^2 \), where \( L \) is the inductance and \( I \) is the current through the inductor. During this phase, the capacitor's energy is at its minimum as all the energy is now in the inductor. This interchange marks the oscillatory nature of LC circuits.

Resonance in Circuits

Resonance occurs in an LC circuit when the inductive and capacitive reactances are equal, resulting in a condition where energy oscillates efficiently between the capacitor and inductor. This balancing happens at the resonance frequency, expressed as \( \omega_0 = \frac{1}{\sqrt{LC}} \).
At resonance, both components share energy equally, ensuring maximum energy transfer. For students: recognizing when a circuit is at resonance can help simplify complex calculations and predict circuit behaviors efficiently.

Oscillating Circuits

Oscillating circuits are those where energy moves back and forth between the capacitor and inductor, creating a predictable cycle. In a freely oscillating LC circuit, these components continue to exchange energy indefinitely in the absence of resistive losses.
The energy flow creates a sinusoidal waveform, illustrating harmonic motion. This concept is crucial for understanding how devices like radios and transmitters function, relying on precise frequency control typical of oscillating circuits.

Energy Distribution in LC Circuits

Energy distribution in an LC circuit differs based on frequency. When driven by an external AC source, the circuit’s frequency impacts how energy is shared between the capacitor and inductor.
By defining \( u = \frac{\overline{U_{C}}}{\overline{U_{C}} + \overline{U_{L}}} \), the energy divide becomes clearer. Before and after resonance, energy share shifts, predominantly favoring the capacitor at lower frequencies and the inductor at higher frequencies. At resonance, the value \( u \) becomes 0.5, showing equal energy partition, which aids in understanding circuit response over a range of frequencies.