Problem 32 - College Physics

Question

A series \(\mathrm{RC}\) circuit has a resistance of \(250 \Omega\) and a capacitance of \(6.0 \mu \mathrm{F}\). If the circuit is driven by a \(60-\mathrm{Hz}\) source, find (a) the capacitive reactance and (b) the impedance of the circuit.

Step-by-Step Solution

Answer

The capacitive reactance is approximately 442.7 \( \Omega \), and the impedance is approximately 509.2 \( \Omega \).

Step 1: Identify Given Values

We are given that the resistance \( R = 250 \Omega \), the capacitance \( C = 6.0 \mu F \) or \( 6.0 \times 10^{-6} F \), and the frequency \( f = 60 \text{ Hz} \).

Step 2: Calculate the Capacitive Reactance

The formula for capacitive reactance is \( X_C = \frac{1}{2\pi f C} \). Substituting the given values, we have:\[X_C = \frac{1}{2\pi (60) (6.0 \times 10^{-6})}\]Calculating this gives:\[X_C \approx \frac{1}{2\pi \times 360 \times 10^{-6}} \approx \frac{1}{2.261 \times 10^{-3}} \approx 442.7 \Omega \]

Step 3: Calculate the Impedance

The impedance \( Z \) of an \( RC \) series circuit is given by:\[Z = \sqrt{R^2 + X_C^2}\]Substituting the resistance and capacitive reactance, we have:\[Z = \sqrt{(250)^2 + (442.7)^2} \]Calculating this gives:\[Z = \sqrt{62500 + 196782.49} = \sqrt{259282.49} \approx 509.2 \Omega \]

Key Concepts

Understanding Capacitive Reactance

Capacitive reactance is a fundamental concept in analyzing AC circuits that include capacitors. It represents the opposition that a capacitor presents to the change in voltage across an AC source. Unlike resistance, which is constant, capacitive reactance decreases with increasing frequency.

Capacitive reactance (\( X_C \)) is given by the formula: \( X_C = \frac{1}{2\pi f C} \), where:

• \( f \) is the frequency of the AC source in Hertz (Hz)
• \( C \) is the capacitance in Farads (F)

A key takeaway here is that as the frequency of the source increases, or the capacitance grows larger, the capacitive reactance reduces, indicating that the circuit can "conduct" the AC signal more easily. For our specific example using \( f = 60 \)Hz and \( C = 6.0 \times 10^{-6} \) F, we calculated a capacitive reactance of approximately \( 442.7 \ Omega \).

Impedance Calculation Made Simple

The concept of impedance is central in AC circuit analysis. It accounts for both resistance and reactance. In a series RC circuit, the total impedance \( Z \) is the combination of the resistance (\( R \)) and the capacitive reactance (\( X_C \)). It is often visualized as the hypotenuse of a right triangle where resistance and reactance form the other two sides.

The formula to calculate impedance is similar to the Pythagorean theorem: \( Z = \sqrt{R^2 + X_C^2} \).

• Here, \( R \) represents the resistive part measured in Ohms (\( \Omega \)).
• \( X_C \) represents the capacitive aspect.

This relationship between resistance and reactance is crucial for understanding how AC currents flow through circuit elements. For instance, with our given values of \( R = 250 \) Ohms and \( X_C \approx 442.7 \) Ohms, we found the total impedance to be approximately \( 509.2 \) Ohms.

The Nature of a Series RC Circuit

A series RC circuit consists of a resistor and capacitor connected end-to-end. Understanding series circuits is fundamental because it lays the groundwork for more complex circuit analysis. In these circuits, the current flowing through the resistor is the same as the current through the capacitor.

Here are some essential characteristics of a series RC circuit:

• Only one path for the current to travel through, meaning the current is the same throughout each component.
• The total voltage across the circuit is the sum of the voltages across the resistor and capacitor.
• It shows how elements like resistance become significant in limiting current flow, while capacitive reactance influences the phase relationship between voltage and current.

These types of circuits are often used in filtering applications and timing circuits due to the relationship between the components. Understanding the role each part plays in the circuit helps in optimizing and predicting how the overall system functions.