An R L C circuit consists of a 48 Ωresistor, a 200 μF capacitor, and an inductor. The rms curren

step 1 in this question we are given an rmc circuit okay so just to draw it up here we have a resistor

An R L C circuit consists of a 48 Ωresistor, a 200 μF...

An $R L C$ circuit consists of a $48 \Omega$ resistor, a $200 \mu \mathrm{F}$ capacitor, and an inductor. The rms current is $2.5 \mathrm{A}$ when the circuit is connected to a $120 \mathrm{V}, 60 \mathrm{Hz}$ outlet. What is the inductance?

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Key Concepts

AC Circuit Impedance

In AC circuits, the total opposition to current flow is represented by a complex impedance that combines both resistance (the real part) and reactance (the imaginary part). This concept allows engineers to use phasor techniques and Ohm's law (V = IZ) in AC analysis, where the magnitude and phase of voltage and current are related to the overall impedance of the circuit.

Reactance of Capacitors and Inductors

Capacitors and inductors exhibit frequency-dependent opposition to current flow known as reactance. Capacitive reactance, given by X_C = 1/(?C), decreases with increasing frequency, while inductive reactance, given by X_L = ?L, increases with frequency. These behaviors cause a phase shift between voltage and current, affecting the overall circuit response.

Resonance in RLC Circuits

Resonance in a series RLC circuit occurs when the inductive and capacitive reactances are equal in magnitude, causing them to cancel each other out. This cancellation results in the circuit’s impedance being purely resistive at the resonant frequency, a condition often used to determine unknown parameters such as inductance or capacitance in the circuit.

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