In an RLC series circuit that includes a source of...

Name: Problem 70, Chapter 21: College Physics
Uploaded: 2019-11-16T20:11:41-08:00
Duration: 3 min 18 s
Channel: Averell Hause
Description: In an RLC series circuit that includes a source of alternating current operating at fixed frequency and voltage, the resistance R is equal to the inductive reactance. If the plate separation of the parallel - plate capacitor is reduced to one - half its original value, the current in the circuit doubles. Find the initial capacitive reactance in terms of R.

In an RLC series circuit that includes a source of alternating current operating at fixed frequency and voltage, the resistance R is equal to the inductive reactance. If the plate separation of the parallel - plate capacitor is reduced to one - half its original value, the current in the circuit doubles. Find the initial capacitive reactance in terms of R.

Key Concepts

RLC Series Circuit

An RLC series circuit consists of a resistor, an inductor, and a capacitor connected in a single loop. The behavior of the circuit under an alternating current (AC) source is determined by the individual impedances of these elements. The overall response, including phase relationships and amplitude of the current, depends on the interplay between resistance, inductive reactance, and capacitive reactance.

Capacitive Reactance

Capacitive reactance is the opposition that a capacitor offers to the change in voltage in an AC circuit. It is inversely proportional to both the frequency of the AC source and the capacitance of the capacitor (X_C = 1/(2?fC)). This means that as the capacitance increases (for example, when the separation between the plates decreases), the capacitive reactance decreases, affecting the overall impedance and current in the circuit.

Inductive Reactance

Inductive reactance is the opposition that an inductor offers to the change in current in an AC circuit. It is directly proportional to the frequency of the AC source and the inductance of the coil (X_L = 2?fL). In circuits where the resistive and inductive components are balanced (for example, when R equals X_L), the phase difference between voltage and current is affected, and this balance plays a critical role in determining the circuit's behavior under different conditions.

Impedance in AC Circuits

Impedance is the combined effect of resistance and reactance (both inductive and capacitive) in an AC circuit. It determines the current flow for a given AC voltage according to the relation I = V/Z, where Z is the total impedance. Changes in the reactance components, such as reducing the capacitive reactance by altering capacitor properties, directly influence the magnitude of the current in the circuit.

Capacitor Geometry

The capacitance of a parallel-plate capacitor is directly proportional to the area of the plates and inversely proportional to the separation between them. When the plate separation is reduced, the capacitance increases, leading to a decrease in the capacitive reactance. This geometric dependence is fundamental in designing circuits where precise control over the capacitance and its reactance is required.

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