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Question Detail
Q factor of the coil is calculated by
Options
Option A: $\omega L$
Option B: $\frac{\omega L}{R}$
Option C: $\frac{\omega L}{C}$
Option D: $\omega LC$
Correct Answer
$\frac{\omega L}{R}$
Solution & Explanation
{"type":"technical","methodBadge":"Concept & Principle","explanation":"The Quality factor (Q-factor) of a coil is a measure of its efficiency in terms of energy storage relative to energy dissipation. It is defined as the ratio of the inductive reactance of the coil to its effective resistance at a given frequency.","workingPrinciple":"In a real inductor, the copper wire has internal resistance $R$. When an AC current flows, the inductor stores energy in the magnetic field ($E_{stored} = \\frac{1}{2}LI^2$) while the resistor dissipates energy as heat ($P_{loss} = I^2R$). The Q-factor is derived as $Q = 2\\pi \\times \\frac{\\text{Maximum energy stored}}{\\text{Energy dissipated per cycle}}$, which simplifies to $\\frac{\\omega L}{R}$ at resonant frequency.","diagramSvg":"RLQ = \\frac{\\omega L}{R}","keyFormulas":["$Q = \\frac{\\omega L}{R}$ тАФ Quality factor of a series RL circuit","$Q = \\frac{1}{R}\\sqrt{\\frac{L}{C}}$ тАФ Quality factor for a series RLC circuit at resonance"],"keyPoints":["A higher Q-factor indicates a lower rate of energy loss and a narrower bandwidth.","The Q-factor is dimensionless.","For a pure inductor (R=0), the Q-factor is theoretically infinite.","Q-factor determines the selectivity of a resonant circuit."],"advantages":["Provides a metric for component 'purity'","Essential for calculating circuit bandwidth ($BW = \\frac{f_r}{Q}$)"],"disadvantages":["Frequency dependent; varies as the operating frequency changes","Affected by skin effect and proximity effect in high-frequency coils"],"applications":["Radio frequency (RF) tuning circuits","Band-pass filter design","Oscillator stability analysis"],"comparisonTable":[],"additionalInfo":["In radio engineering, coils with high Q are preferred for sharp tuning.","Option A is simply inductive reactance ($X_L$). Option C is the ratio of reactance to capacitance (incorrect unit). Option D represents the product of reactance and capacitance."],"answer":"B is correct тАФ The Q factor is defined as the ratio of inductive reactance $(\\omega L)$ to the series resistance $(R)$ of the coil.","correctedOptions":["$\\omega L$","$\\frac{\\omega L}{R}$","$\\frac{\\omega L}{C}$","$\\omega LC$"],"coreConcepts":["Inductive Reactance","Energy Dissipation","Resonance in RLC Circuits"],"crossTopicTip":"Remember that for parallel circuits, the Q-factor is the reciprocal of the series case: $Q = \\frac{R}{\\omega L}$."}
