The Question is from neet-physics alternating-current choke-coil-transformer, and it was created by the author P.NARASIMHA

Question

A choke coil and a resistor are connected in series to an AC source. If the inductance of the choke coil is $L$ and the resistance is $R$, under what condition will the power consumed by the circuit be independent of the frequency of the AC source?

P.NARASIMHA · Accepted Answer

## CORE INFORMATION ✓ **Answer**: C ## CONCEPT FRAMEWORK **Primary Concept**: AC Circuits, Power in RLC circuits **Related Topics**: Impedance, Reactance, Resonance **Prerequisites**: Understanding of AC circuit analysis, Impedance, and Power calculations. ## SOLUTION PATHWAY **Given Information**: * A choke coil (inductor) and a resistor are in series. * Inductance $L$ and resistance $R$. * AC source. **Method & Steps**: 1. **Impedance of the circuit**: The impedance $Z$ of the series RL circuit is given by $Z = \sqrt{R^2 + (\omega L)^2}$, where $\omega$ is the angular frequency of the AC source. Reasoning: The impedance is the combined effect of resistance and inductive reactance in the circuit. Formula/Rule: $Z = \sqrt{R^2 + X_L^2}$, where $X_L = \omega L$ is the inductive reactance. 2. **Power consumed**: The power consumed by the circuit is given by $P = I^2 R = \frac{V^2 R}{Z^2}$, where $I$ is the current and $V$ is the source voltage. Working: Substituting $Z$, we have $P = \frac{V^2 R}{R^2 + (\omega L)^2}$. Key Point: The power consumed depends on the frequency $\omega$ through the impedance term. 3. **Frequency independence**: For the power to be independent of frequency, the term $(\omega L)^2$ should be negligible compared to $R^2$. Result: This happens when $R^2 >> (\omega L)^2$ or $R >> \omega L$. Verification: If $R >> \omega L$, then $Z \approx R$, and $P \approx \frac{V^2}{R}$, which is independent of $\omega$. ## OPTION ANALYSIS [A] R << ωL :x: In this case, the inductive reactance dominates, and the power becomes highly frequency-dependent. [B] R = ωL :x: The power will still be frequency-dependent, although less so than in option A. [C] R >> ωL ✓: This is the correct condition. The resistance dominates, making the power essentially independent of frequency. [D] R = 0 :x: If R = 0, the circuit becomes purely inductive, and no real power is dissipated. The power expression becomes undefined. ## LEARNING AIDS **Common Mistakes**: 1. Confusing impedance with resistance: Impedance includes both resistance and reactance. Remember to consider both when calculating power in AC circuits. 2. Neglecting the frequency dependence of reactance: Reactance is directly proportional to frequency. This is the key reason for the frequency dependence of power in this circuit. **Quick Tips**: 1. When $R >> \omega L$, the circuit behaves almost like a purely resistive circuit. 2. When $R << \omega L$, the circuit behaves almost like a purely inductive circuit.

P.NARASIMHA · Answer

R << ωL

P.NARASIMHA · Answer

R = ωL

P.NARASIMHA · Answer

The correct Option is "R >> ωL"

P.NARASIMHA · Answer

R = 0

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