A coil of resistance 35.0Ωand inductance 20.5 H is in series with a capacitor and a 200-V (rms),

step 1 We have the reactance for the inductor is 2 pi fl which will come out as 1 .29 times 10 to the 4

A coil of resistance 35.0Ωand inductance 20.5 H is in series...

A coil of resistance 35.0$\Omega$ and inductance 20.5 $\mathrm{H}$ is in series with a capacitor and a $200-\mathrm{V}$ (rms), $100-\mathrm{Hz}$ source. The rms current in the circuit is 4.00 $\mathrm{A}$ . (a) Calculate the capacitance in the circuit. (b) What is $\Delta V_{\mathrm{rms}}$ across the coil?

Key Concepts

RLC Series Circuit Analysis

A series RLC circuit is one in which a resistor, an inductor, and a capacitor are connected in a single loop. The overall behavior of the circuit is governed by the frequency-dependent impedances of the inductor and capacitor, combined with the fixed resistance. Understanding these circuits requires analyzing how the inductive and capacitive reactances interact, possibly cancelling each other out, which affects both the amplitude and phase of the total impedance.

Impedance in AC Circuits

Impedance is the measure of opposition that a circuit presents to the flow of alternating current (AC). It extends the concept of resistance to AC circuits by incorporating both the resistive (real) and reactive (imaginary) components. In series circuits, the total impedance is calculated using the Pythagorean sum of the resistance and the net reactance, which determines the magnitude of the current for a given source voltage.

Inductive and Capacitive Reactance

Inductive reactance, given by ?L, and capacitive reactance, given by 1/(?C), describe how inductors and capacitors resist changes in current in AC circuits. These reactances depend on the frequency of the AC source, with inductive reactance increasing with frequency and capacitive reactance decreasing with frequency. The net reactance, which is the difference between inductive and capacitive reactances, is crucial for determining the circuit’s overall impedance and its phase behavior.

RMS Values in AC Circuit Analysis

Root mean square (RMS) values provide a measure of the effective voltage or current in AC circuits. They represent the equivalent DC values that would deliver the same power to a resistive load. Using RMS values simplifies the analysis of AC circuits by allowing the application of Ohm’s Law and power calculations in a manner analogous to DC circuits.

Voltage Division in AC Circuits

Voltage division in AC circuits refers to the method of determining the voltage across a particular component by considering its impedance relative to the total impedance of the series circuit. This concept, similar to the voltage divider rule in DC circuits, takes into account both resistive and reactive components, allowing for the calculation of individual voltage drops even in circuits where phase differences are significant.

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