Show that the fraction of electromagnetic energy lost (to thermal energy) per cycle in a lightly damped (R² ≪ 4L/C) LRC circuit is approximately ΔUU=2πRLω=2πQ\frac{\Delta U}{U}=\frac{2\pi R}{L\omega}=\frac{2\pi}{Q}. The quantity Q can be defined as Q = Lω/R, and is called the Q-value, or quality factor, of the circuit and is a measure of the damping present. A high Q-value means smaller damping and less energy input required to maintain oscillations.

Ch. 30 - Inductance, Electromagnetic Oscillations, and AC Circuits

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 30 - Inductance, Electromagnetic Oscillations, and AC Circuits

Problem 83

Chapter 29, Problem 83

Show that the fraction of electromagnetic energy lost (to thermal energy) per cycle in a lightly damped (R² ≪ 4L/C) LRC circuit is approximately $\frac{Δ U}{U} = \frac{2 π R}{L ω} = \frac{2 π}{Q}$ . The quantity Q can be defined as Q = Lω/R, and is called the Q-value, or quality factor, of the circuit and is a measure of the damping present. A high Q-value means smaller damping and less energy input required to maintain oscillations.

Verified step by step guidance

Start by understanding the energy loss in an LRC circuit. The energy stored in the circuit oscillates between the inductor and capacitor, but due to the resistance (R), some energy is dissipated as thermal energy in each cycle.

The energy stored in the circuit at any given time is proportional to the square of the amplitude of the oscillating current or voltage. Let the total energy in the circuit be denoted as U. The energy lost per cycle, ΔU, is proportional to the resistance R and the current squared.

The Q-value (quality factor) is defined as $ Q = \frac{L \omega}{R} $, where $ L $ is the inductance, $ \omega $ is the angular frequency, and $ R $ is the resistance. A high Q-value corresponds to low damping and minimal energy loss per cycle.

The fraction of energy lost per cycle can be expressed as $ \frac{\Delta U}{U} $. Using the relationship between Q, R, and $ \omega $, and considering the lightly damped condition $ R^2 \ll \frac{4L}{C} $, we can derive that $ \frac{\Delta U}{U} \approx \frac{2\pi}{Q} $.

Substitute $ Q = \frac{L \omega}{R} $ into the expression for $ \frac{\Delta U}{U} $, and simplify to show that $ \Delta U \approx 2\pi R \cdot U \cdot \frac{1}{L \omega} $. This confirms the relationship between energy loss, resistance, and the Q-value in a lightly damped LRC circuit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

LRC Circuit

An LRC circuit consists of an inductor (L), a resistor (R), and a capacitor (C) connected in series or parallel. It exhibits oscillatory behavior due to the energy exchange between the inductor's magnetic field and the capacitor's electric field. The circuit's dynamics are governed by differential equations that describe how voltage and current change over time, leading to phenomena such as resonance and damping.

Quality Factor (Q-value)

The quality factor, or Q-value, is a dimensionless parameter that quantifies the damping of an oscillating system, particularly in LRC circuits. It is defined as the ratio of the stored energy to the energy lost per cycle. A high Q-value indicates low energy loss and sharp resonance, while a low Q-value signifies higher damping and broader resonance peaks, affecting the circuit's performance in applications like filters and oscillators.

Damping and Energy Loss

Damping refers to the reduction of amplitude in oscillatory systems due to energy loss, often manifested as thermal energy in resistive components. In lightly damped systems, such as those with R² ≪ 4L/C, the energy loss per cycle is small compared to the total energy stored, allowing sustained oscillations. Understanding the relationship between resistance, inductance, and capacitance is crucial for analyzing how energy is dissipated in these circuits.

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